Basic representation theory[ draft ]

Incomplete notes for a mini-course given at UCAS Summer 2017

Wen-Wei Li

Version December 10 2018

Contents1 Topological groups 2

11 Definition of topological groups 212 Local fields 413 Measures and integrals 514 Haar measures 615 The modulus character 916 Interlude on analytic manifolds 1117 More examples 1318 Homogeneous spaces 14

2 Convolution 1721 Convolution of functions 1722 Convolution of measures 19

3 Continuous representations 2031 General representations 2032 Matrix coefficients 2333 Vector-valued integrals 2534 Action of convolution algebra 2635 Smooth vectors archimedean case 28

4 Unitary representation theory 2841 Generalities 2942 External tensor products 3243 Square-integrable representations 3344 Spectral decomposition the discrete part 3745 Usage of compact operators 3946 The case of compact groups 4047 Basic character theory 41

5 Unitary representation of compact Lie groups (a sketch) 4251 Review of basic notions 4352 Algebraic preparations 4453 Weyl character formula semisimple case 4754 Extension to non-semisimple groups 50

1

References 50

1 Topological groups11 Definition of topological groupsFor any group 119866 we write 1 = 1119866 for its identity element and write 119866op for its opposite group in otherwords 119866op has the same underlying set as 119866 but with the new multiplication (119909 119910) ↦ 119910119909

Denote by 1201381113568 the group 119911 isin ℂtimes ∶ |119911| = 1 endowed with its usual topological structure

Definition 11 A topological group is a topological space 119866 endowed with a group structure such thatthe maps (119909 119910) ↦ 119909119910 and 119909 ↦ 119909minus1113568 are continuous Unless otherwise specified we always assume that119866 is Hausdorff

By a homomorphism 120593 ∶ 1198661113568 rarr 1198661113569 between topological groups we mean a continuous homomor-phism This turns the collection of all topological groups into a category TopGrp and it makes senseto talk about isomorphisms etc We write Hom(1198661113568 1198661113569) and Aut(119866) for the sets of homomorphisms andautomorphisms in TopGrp

Let 119866 be a topological group As is easily checked (i) 119866op is also a topological group (ii) inv ∶ 119909 ↦119909minus1113568 is an isomorphism of topological groups 119866 simminusrarr 119866op with inv ∘ inv = id and (iii) for every 119892 isin 119866the translation map 119871119892 ∶ 119909 ↦ 119892119909 (resp 119877119892 ∶ 119909 ↦ 119909119892) is a homeomorphism from 119866 to itself indeed119871119909minus1113578119871119909 = id = 119877119909minus1113578119877119909

By translating we see that if 1199771113568 is the set of open neighborhoods of 1 then the set of open neigh-borhoods of 119909 isin 119866 equals 119909119880 ∶ 119880 isin 1199771113568 it also equals 119880119909 ∶ 119880 isin 1199771113568

We need a few elementary properties of topological groups Notation for subsets 119860119861 sub 119866 wewrite 119860119861 = 119886119887 ∶ 119886 isin 119860 119887 isin 119861 sub 119866 similarly for 119860119861119862 and so forth Also put 119860minus1113568 = 119886minus1113568 ∶ 119886 isin 119860

Proposition 12 For any topological group 119866 (not presumed Hausdorff) the following are equivalent

(i) 119866 is Hausdorff

(ii) the intersection of all open neighborhoods containing 1 is 1

(iii) 1 is closed in 119866

Proof It is clear that (i) ⟹ (ii) (iii) Also (ii) ⟺ (iii) since it is routine to check that 1 = ⋂119880ni1113568119880 Let us prove (ii) ⟹ (i) as follows Let 120584 ∶ 119866 times 119866 rarr 119866 be the function (119909 119910) ↦ 119909119910minus1113568 Then 119866 is

Hausforff if and only if the diagonal Δ119866 sub 119866 times 119866 is closed whilst Δ119866 = 120584minus1113568(1) Now (ii) implies (iii)which in turn implies Δ119866 is closed

Proposition 13 Let 119866 be a topological group (not necessarily Hausdorff) and 119860119861 sub 119866 be subsets

1 If one of 119860 119861 is open then 119860119861 is open

2 If both 119860 119861 are compact then so is 119860119861

3 If one of 119860 119861 is compact and the other is closed then 119860119861 is closed

Proof Suppose that 119860 is open Then 119860119861 = ⋃119887isin119861119860119887 is open as wellSuppose that 119860 119861 are compact The image 119860119861 of 119860 times 119861 under multiplication is also compactFinally suppose 119860 is compact and 119861 is closed If 119909 notin 119860119861 then 119909119861minus1113568 is closed and disjoint from

119860 If we can find an open subset 119880 ni 1 such that 119880119860 cap 119909119861minus1113568 = empty then 119880minus1113568119909 ni 119909 will be an openneighborhood disjoint from 119860119861 proving that 119860119861 is closed

2

To find 119880 set 119881 ∶= 119866 ∖ 119909119861minus1113568 For every 119886 isin 119860 sub 119881 there exists an open subset 119880 prime119886 ni 1 with

119880 prime119886119886 sub 119881 Take an open 119880119886 ni 1 with 119880119886119880119886 sub 119880 prime

119886 Compactness furnishes a finite subset 1198601113567 sub 119860 suchthat 119860 sub ⋃119905isin1198601113577 119880119905119905 We claim that 119880 ∶= ⋂119905isin1198601113577 119880119905 satisfies the requirements Indeed 119880 ni 1 and each119886 isin 119860 belongs to 119880119905119905 for some 119905 isin 1198601113567 hence

119880119886 sub 119880119880119905119905 sub 119880119905119880119905119905 sub 119880 prime119905 119905 sub 119881

thus 119880119860 sub 119881

Recall that a topological space is locally compact if every point has a compact neighborhood

Proposition 14 Let 119867 be a subgroup of a topological group 119866 (not necessarily Hausdorff) Endow119866119867 with the quotient topology

1 The quotient map 120587 ∶ 119866 rarr 119866119867 is open and continuous

2 If 119866 is locally compact so is 119866119867

3 119866119867 is Hausdorff (resp discrete) if and only if 119867 is closed (resp open)

The same holds for 119867119866

Proof The quotient map is always continuous If119880 sub 119866 is open then 120587minus1113568(120587(119880)) = 119880119867 is open by 13hence 120587(119880) is open

Suppose 119866 is locally compact By homogeneity it suffices to argue that the coset 119867 = 120587(1) hascompact neighborhood in 119866119867 Let 119870 ni 1 be a compact neighborhood in 119866 Choose a neighborhood119880 ni 1 such that 119880minus1113568119880 sub 119870 Claim 120587(119880) sub 120587(119870) Indeed if 119892119867 isin 120587(119880) then the neighborhood 119880119892119867intersects 120587(119880) that is 119906119892119867 = 119906prime119867 for some 119906 119906prime isin 119880 Hence 119892119867 = 119906minus1113568119906prime119867 isin 120587(119880minus1113568119880) sub 120587(119870)Therefore 120587(119880) ni 120587(1) is an open neighborhood with 120587(119880) compact since 120587(119870) is compact

If 119866119867 is Hausdorff (resp discrete) then 119867 = 120587minus1113568(120587(1)) is closed (resp open) Conversely supposethat 119867 is closed in 119866 Given cosets 119909119867 ne 119910119867 choose an open neighborhood 119881 ni 1 in 119866 such that119881119909cap 119910119867 = empty equivalently 119881119909119867 cap 119910119867 = empty Then choose an open 119880 ni 1 in 119866 such that 119880minus1113568119880 sub 119881 Itfollows that 119880119909119867 cap 119880119910119867 = empty and these are disjoint open neighborhoods of 120587(119909) and 120587(119910) All in all119866119867 is Hausdorff

On the other hand 119867 is open implies 120587(119867) is open thus all singletons in 119866119867 are open whence thediscreteness

As for 119867119866 we pass to 119866op

Lemma 15 Let119866 be a locally compact group (not necessarily Hausdorff) and let119867 sub 119866 be a subgroupEvery compact subset of 119867119866 is the image of some compact subset of 119866

Proof Let 119880 ni 1 be an open neighborhood in 119866 with compact closure 119880 For every compact subset 119870of 119867119866 we have an open covering 119870 sub ⋃119892isin119866 120587(119892119880) hence there exists 1198921113568 hellip 119892119899 isin 119866 such that

119870 sub1198991113996119894=1113568

120587(119892119894119880) = 120587

⎛⎜⎜⎜⎜⎜⎝1198991113996119894=1113568

119892119894119880

⎞⎟⎟⎟⎟⎟⎠ sub 120587

⎛⎜⎜⎜⎜⎜⎝1198991113996119894=1113568

119892119894

⎞⎟⎟⎟⎟⎟⎠

Take the compact subset 119870 ∶= ⋃119899119894=1113568 119892119894 cap 120587minus1113568(119870) and observe that 120587(119870) = 119870

Remark 16 The following operations on topological groups are evident

1 Let 119867 sub 119866 be a closed normal subgroup then 119866119867 is a topological group by Proposition 14locally compact if 119866 is We leave it to the reader to characterize 119866119867 by universal properties inTopGrp

3

2 If 1198661113568 1198661113569 are topological groups then 1198661113568 times 1198661113569 is naturally a topological group Again it hasan outright categorical characterization the product in TopGrp More generally one can formfibered products in TopGrp

3 In a similar vein limlarrminusminus exist in TopGrp their underlying abstract groups are just the usual limlarrminusminus

Harmonic analysis in its classical sense applies mainly to locally compact topological groups Here-after we adopt the shorthand locally compact groupsRemark 17 The family of locally compact groups is closed under passing to closed subgroups Hausdorffquotients and finite direct products However infinite direct products usually yield non-locally compactgroups This is one of the motivation for introducing restrict products into harmonic analysis

Example 18 Discrete groups are locally compact finite groups are compact

Example 19 The familiar groups (ℝ +) (ℂ +) are locally compact so are (ℂtimes sdot) and (ℝtimes sdot) Theidentity connected component ℝtimes

gt1113567 of ℝtimes is isomorphic to (ℝ +) through the logarithm The quotientgroup (ℝℤ +) ≃ 1201381113568 (via 119911 ↦ 1198901113569120587119894119911) is compact

12 Local fieldsJust as the case of groups a locally compact field is a field 119865 with a locally compact Hausdorff topologysuch that (119865 +) is a topological group and that (119909 119910) ↦ 119909119910 and 119909 ↦ 119909minus1113568 (on 119865times) are both continuous

Definition 110 A local field is a locally compact field that is not discrete

A detailed account of local fields can be found in any textbook on algebraic number theory Thetopology on a local field 119865 is always induced by an absolute value | sdot |119865 ∶ 119865 rarr ℝge1113567 satisfying

|119909|119865 = 0 ⟺ 119909 = 0 |1|119865 = 1|119909 + 119910|119865 le |119909|119865 + |119910|119865|119909119910|119865 = |119909| sdot |119910|119865

Furthermore 119865 is complete with respect to | sdot | Up to continuous isomorphisms local fields are classifiedas follows

Archimedean The fields ℝ and ℂ equipped with the usual absolute values

Non-archimedean characteristic zero The fieldsℚ119901 = ℤ119901[1113568119901 ] (the 119901-adic numbers where 119901 is a prime

number) or their finite extensions

Non-archimedean characteristic 119901 gt 0 The fields 120125119902((119905)) = 120125119902J119905K[ 1113568119905 ] of Laurent series in the variable119905 or their finite extensions where 119902 is a power of 119901 Here 120125119902 denotes the finite field of 119902 elements

Let 119865 be a non-archimedean local field It turns out the ultrametric inequality is satisfied

|119909 + 119910|119865 le max|119909|119865 |119910|119865 with equality when |119909| ne |119910|

Furthermore 120108119865 = 119909 ∶ |119909| le 1 is a subring called the ring of integers and 120109119865 = 119909 ∶ |119909| lt 1 is itsunique maximal ideal In fact 120109119865 is of the form (120603119865) here 120603119865 is called a uniformizer of 119865 and

119865times = 120603ℤ119865 times 120108times119865 120108times119865 = 119909 ∶ |119909| = 1

The normalized absolute value | sdot |119865 for non-archimedean 119865 is defined by

|120603119865 | = 119902minus1113568 119902 ∶= |120108119865120109119865 |

4

where 120603119865 is any uniformizer We will interpret | sdot |119865 in terms of modulus characters in 133If 119864 is a finite extension of any local field 119865 then 119864 is also local and | sdot |119865 admits a unique extension

to 119864 given by| sdot |119864 = |119873119864119865(sdot)|

1113568[119864∶119865]119865

where 119873119864119865 ∶ 119864 rarr 119865 is the norm map It defines the normalized absolute value on 119864 in the non-archimedean caseRemark 111 The example ℝ (resp ℚ119901) above is obtained by completing ℚ with respect to the usualabsolute value (resp the 119901-adic one |119909|119901 = 119901minus119907119901(119909) where 119907119901(119909) = 119896 if 119909 = 119901119896 119906119907 ne 0 with 119906 119907 isin ℤ coprimeto 119901 and 119907119901(0) = +infin)

Likewise 120125119902((119905)) is the completion of the function field120125119902(119905)with respect to |119909|1113567 = 119902minus1199071113577(119909) where 1199071113567(119909)is the vanishing order of the rational function 119909 at 0

In both cases the local field 119865 arises from completing some global fieldℚ or 120125119902(119905) or more generallytheir finite extensions The adjective ldquoglobalrdquo comes from geometry which is manifest in the case 120125119902(119905)it is the function field of the curveℙ1113568120125119902 and120125119902((119905)) should be thought as the function field on the ldquopuncturedformal diskrdquo at 0

13 Measures and integralsWe shall only consider Radon measures on a locally compact Hausdorff space 119883 These measures areby definition Borel locally finite and inner regular

Define the ℂ-vector space

119862119888(119883) ∶= 1114106119891 ∶ 119883 rarr ℂ continuous compactly supported1114109

= 1113996119870sub119883

compact

119862119888(119883 119870) 119862119888(119883 119870) ∶= 119891 isin 119862119888(119883) ∶ Supp(119891) sub 119870

We write 119862119888(119883)+ ∶= 1114106119891 isin 119862119888(119883) ∶ 119891 ge 01114109Remark 112 Following Bourbaki we identify positive Radon measures 120583 on 119883 and positive linearfunctionals 119868 = 119868120583 ∶ 119862119888(119883) rarr ℂ This means that 119868(119891) ge 0 if 119891 isin 119862119888(119883)+ In terms of integrals119868120583(119891) = int119883 119891 d120583 This is essentially a consequence of the Riesz representation theorem

Furthermore to prescribe a positive linear functional 119868 is the same as giving a function 119868 ∶ 119862119888(119883)+ rarrℝge1113567 that satisfies

bull 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)

bull 119868(119905119891) = 119905119868(119891) when 119905 isin ℝge1113567

To see this write 119891 = 119906 + 119894119907 isin 119862119888(119883) where 119906 119907 ∶ 119883 rarr ℝ furthermore any real-valued 119891 isin 119862119888(119883) canbe written as 119891 = 119891+ minus 119891minus as usual where 119891plusmn isin 119862119888(119883)+ All these decompositions are canonicalRemark 113 The complex Radon measures correspond to linear functionals 119868 ∶ 119862119888(119862) rarr ℂ such that119868 |119862119888(119883119870) is continuous with respect to sdot infin119870 ∶= sup119870 | sdot | for all 119870 In other words 119868 is continuous forthe topology of limminusminusrarr119870

119862119888(119883 119870) ≃ 119862119888(119883)

Now let119866 be a locally compact group and suppose that119883 is endowed with a continuous left119866-actionContinuity here means that the action map

119886 ∶ 119866 times 119883 ⟶ 119883(119892 119909)⟼ 119892119909

5

is continuous Similarly for right 119866-actions This action transports the functions 119891 isin 119862119888(119883) as well

119891119892 ∶= 1114094119909 ↦ 119891(119892119909)1114097 left action119892119891 ∶= 1114094119909 ↦ 119891(119909119892)]1114097 right action

These terminologies are justified as 119891119892119892prime = 11141001198911198921114103119892prime

and 119892119892prime119891 = 119892 1114100119892prime1198911114103 they also preserve 119862119888(119883)+ There-fore for 119866 acting on the left (resp on the right) of 119883 it also acts from the same side on the space ofpositive Radon measures on 119883 by transport of structure In terms of positive linear functionals

119868 ⟼ 1114094119892119868 ∶ 119891 ↦ 119868(119891119892)1114097 left action

119868 ⟼ 1114094119868119892 ∶ 119891 ↦ 119868(119892119891)1114097 right action

This pair of definitions is swapped under 119866 119866opBy taking transposes 119866 also transports complex Radon measures (Remark 113) Mnemonic tech-

niqued(119892120583)(119909) = d120583(119892minus1113568119909) d(120583119892)(119909) = d120583(119909119892minus1113568)

since a familiar change of variables yields

1114009119883119891(119909) d(119892120583)(119909) transpose= 119868(119891119892) = 1114009

119883119891(119892119909) d120583(119909) = 1114009

119883119891(119909) d120583(119892minus1113568119909)

1114009119883119891(119909) d(120583119892)(119909) transpose= 119868(119892119891) = 1114009

119883119891(119909119892) d120583(119909) = 1114009

119883119891(119909) d120583(119909119892minus1113568)

and 119891 isin 119862119888(119883) is arbitrary These formulas extend to all 119891 isin 1198711113568(119883 120583) by approximation

Definition 114 Suppose that 119866 acts continuous on the left (resp right) of 119883 Let 120594 ∶ 119866 rarr ℝtimesgt1113567

by a continuous homomorphism We say that a complex Radon measure 120583 is quasi-invariant or aneigenmeasure with eigencharacter 120594 if 119892120583 = 120594(119892)minus1113568120583 (resp 120583119892 = 120594(119892)minus1113568120583) for all 119892 isin 119866 This can alsobe expressed as

d120583(119892119909) = 120594(119892) d120583(119909) left actiond120583(119909119892) = 120594(119892) d120583(119909) right action

When 120594 = 1 we call 120583 an invariant measure

Observe that 120583 is quasi-invariant with eigencharacter 120594 under left119866-action if and only if it is so underthe right 119866op-actionRemark 115 Let 120594 120578 be continuous homomorphisms 119866 rarr ℂtimes Suppose that 119891 ∶ 119883 rarr ℂ is contin-uous with 119891(119892119909) = 120578(119892)119891(119909) (resp 119891(119909119892) = 120578(119892)119891(119909)) for all 119892 and 119909 Then 120583 is quasi-invariant witheigencharacter 120594 if and only if 119891120583 is quasi-invariant with eigencharacter 120578120594

In particular we may let 119866 act on 119883 = 119866 by left (resp right) translations Therefore it makes senseto talk about left and right invariant measures on 119866

14 Haar measuresIn what follows measures are always nontrivial positive Radon measures

Definition 116 Let 119866 be a locally compact group A left (resp right) invariant measure on 119866 is calleda left (resp right) Haar measure

6

The group ℝtimesgt1113567 acts on the set of left (resp right) Haar measures by rescaling For commutative

groups we make no distinction of left and right

Example 117 If 119866 is discrete the counting measure Count(119864) = |119864| is a left and right Haar measureWhen 119866 is finite it is customary to take the normalized version 120583 ∶= |119866|minus1113568Count

Definition 118 Let 119866 be a locally compact For 119891 isin 119862119888(119866) we write 119891 ∶ 119909 ↦ 119891(119909minus1113568) 119891 isin 119862119888(119866)For any Radon measure 120583 on 119866 let be the Radon measure with d(119909) = d120583(119909minus1113568) in terms of linearfunctionals 119868(119891) = 119868120583( 119891)

Lemma 119 In the situation above 120583 is a left (resp right) Haar measure if and only if is a right(resp left) Haar measure

Proof An instance of transport of structure because 119909 ↦ 119909minus1113568 is an isomorphism of locally compactgroups 119866 simminusrarr 119866op

Theorem 120 (A Weil) For every locally compact group 119866 there exists a left (resp right) Haar mea-sure on 119866 They are unique up to ℝtimes

gt1113567-action

Proof The following arguments are taken from Bourbaki [1 VII sect12] Upon replacing 119866 by 119866op itsuffices to consider the case of left Haar measures For the existence part we seek a left 119866-invariantpositive linear functional on 119862119888(119866)+ Write

119862119888(119866)lowast+ ∶= 119862119888(119866)+ ∖ 0

For any compact subset 119870 sub 119866 define 119862119888(119866 119870)+ and 119862119888(119866 119870)lowast+ by intersecting 119862119888(119866 119870) with 119862119888(119866)+and 119862119888(119866)lowast+ Observe that for all 119891 isin 119862119888(119866)+ and 119892 isin 119862119888(119866)lowast+ there exist 119899 ge 1 1198881113568 hellip 119888119899 isin ℝge1113567 and1199041113568 hellip 119904119899 isin 119866 such that

119891 le 11988811135681198921199041113578 + hellip119888119899119892119904119899 119892119904119894(119909) ∶= 119892(119904119894119909)Indeed there is an open subset 119880 sub 119866 such that inf119880 119892 gt 0 now cover Supp(119891) by finitely many1199041113568119880hellip 119904119899119880

Given 119891 119892 as before define (119891 ∶ 119892) to be the infimum of 1198881113568+hellip+119888119899 among all choices of (1198881113568 hellip 1199041113568 hellip)satisfying the bound above We contend that

(119891119904 ∶ 119892) = (119891 ∶ 119892) 119904 isin 119866 (11)(119905119891 ∶ 119892) = 119905(119891 ∶ 119892) 119905 isin ℝge1113567 (12)

(1198911113568 + 1198911113569 ∶ 119892) le (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) (13)

(119891 ∶ 119892) ge sup 119891sup 119892 (14)

(119891 ∶ ℎ) le (119891 ∶ 119892)(119892 ∶ ℎ) ℎ 119892 isin 119862119888(119866)lowast+ (15)

0 lt 1(1198911113567 ∶ 119891)

le (119891 ∶ 119892)(1198911113567 ∶ 119892)

le (119891 ∶ 1198911113567) 119891 1198911113567 119892 isin 119862119888(119866)lowast+ (16)

and that for all 1198911113568 1198911113569 ℎ isin 119862119888(119866)+ with ℎ|Supp(1198911113578+1198911113579) ge 1 and all 120598 gt 0 there is a compact neighborhood119870 ni 1 such that

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1198911113568 + 1198911113569 ∶ 119892) + 120598(ℎ ∶ 119892) 119892 isin 119862119888(119866 119870)lowast+ (17)

The properties (11) mdash (14) are straightforward For (15) note that

119891 le 1114012119894119888119894119892119904119894 119892 le 1114012

119895119889119895ℎ119905119895 ⟹ 119891 le1114012

119894119895119888119894119889119895ℎ119905119895119904119894

7

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

References 50

1 Topological groups11 Definition of topological groupsFor any group 119866 we write 1 = 1119866 for its identity element and write 119866op for its opposite group in otherwords 119866op has the same underlying set as 119866 but with the new multiplication (119909 119910) ↦ 119910119909

Denote by 1201381113568 the group 119911 isin ℂtimes ∶ |119911| = 1 endowed with its usual topological structure

Definition 11 A topological group is a topological space 119866 endowed with a group structure such thatthe maps (119909 119910) ↦ 119909119910 and 119909 ↦ 119909minus1113568 are continuous Unless otherwise specified we always assume that119866 is Hausdorff

By a homomorphism 120593 ∶ 1198661113568 rarr 1198661113569 between topological groups we mean a continuous homomor-phism This turns the collection of all topological groups into a category TopGrp and it makes senseto talk about isomorphisms etc We write Hom(1198661113568 1198661113569) and Aut(119866) for the sets of homomorphisms andautomorphisms in TopGrp

Let 119866 be a topological group As is easily checked (i) 119866op is also a topological group (ii) inv ∶ 119909 ↦119909minus1113568 is an isomorphism of topological groups 119866 simminusrarr 119866op with inv ∘ inv = id and (iii) for every 119892 isin 119866the translation map 119871119892 ∶ 119909 ↦ 119892119909 (resp 119877119892 ∶ 119909 ↦ 119909119892) is a homeomorphism from 119866 to itself indeed119871119909minus1113578119871119909 = id = 119877119909minus1113578119877119909

By translating we see that if 1199771113568 is the set of open neighborhoods of 1 then the set of open neigh-borhoods of 119909 isin 119866 equals 119909119880 ∶ 119880 isin 1199771113568 it also equals 119880119909 ∶ 119880 isin 1199771113568

We need a few elementary properties of topological groups Notation for subsets 119860119861 sub 119866 wewrite 119860119861 = 119886119887 ∶ 119886 isin 119860 119887 isin 119861 sub 119866 similarly for 119860119861119862 and so forth Also put 119860minus1113568 = 119886minus1113568 ∶ 119886 isin 119860

Proposition 12 For any topological group 119866 (not presumed Hausdorff) the following are equivalent

(i) 119866 is Hausdorff

(ii) the intersection of all open neighborhoods containing 1 is 1

(iii) 1 is closed in 119866

Proof It is clear that (i) ⟹ (ii) (iii) Also (ii) ⟺ (iii) since it is routine to check that 1 = ⋂119880ni1113568119880 Let us prove (ii) ⟹ (i) as follows Let 120584 ∶ 119866 times 119866 rarr 119866 be the function (119909 119910) ↦ 119909119910minus1113568 Then 119866 is

Hausforff if and only if the diagonal Δ119866 sub 119866 times 119866 is closed whilst Δ119866 = 120584minus1113568(1) Now (ii) implies (iii)which in turn implies Δ119866 is closed

Proposition 13 Let 119866 be a topological group (not necessarily Hausdorff) and 119860119861 sub 119866 be subsets

1 If one of 119860 119861 is open then 119860119861 is open

2 If both 119860 119861 are compact then so is 119860119861

3 If one of 119860 119861 is compact and the other is closed then 119860119861 is closed

Proof Suppose that 119860 is open Then 119860119861 = ⋃119887isin119861119860119887 is open as wellSuppose that 119860 119861 are compact The image 119860119861 of 119860 times 119861 under multiplication is also compactFinally suppose 119860 is compact and 119861 is closed If 119909 notin 119860119861 then 119909119861minus1113568 is closed and disjoint from

119860 If we can find an open subset 119880 ni 1 such that 119880119860 cap 119909119861minus1113568 = empty then 119880minus1113568119909 ni 119909 will be an openneighborhood disjoint from 119860119861 proving that 119860119861 is closed

2

To find 119880 set 119881 ∶= 119866 ∖ 119909119861minus1113568 For every 119886 isin 119860 sub 119881 there exists an open subset 119880 prime119886 ni 1 with

119880 prime119886119886 sub 119881 Take an open 119880119886 ni 1 with 119880119886119880119886 sub 119880 prime

119886 Compactness furnishes a finite subset 1198601113567 sub 119860 suchthat 119860 sub ⋃119905isin1198601113577 119880119905119905 We claim that 119880 ∶= ⋂119905isin1198601113577 119880119905 satisfies the requirements Indeed 119880 ni 1 and each119886 isin 119860 belongs to 119880119905119905 for some 119905 isin 1198601113567 hence

119880119886 sub 119880119880119905119905 sub 119880119905119880119905119905 sub 119880 prime119905 119905 sub 119881

thus 119880119860 sub 119881

Recall that a topological space is locally compact if every point has a compact neighborhood

Proposition 14 Let 119867 be a subgroup of a topological group 119866 (not necessarily Hausdorff) Endow119866119867 with the quotient topology

1 The quotient map 120587 ∶ 119866 rarr 119866119867 is open and continuous

2 If 119866 is locally compact so is 119866119867

3 119866119867 is Hausdorff (resp discrete) if and only if 119867 is closed (resp open)

The same holds for 119867119866

Proof The quotient map is always continuous If119880 sub 119866 is open then 120587minus1113568(120587(119880)) = 119880119867 is open by 13hence 120587(119880) is open

Suppose 119866 is locally compact By homogeneity it suffices to argue that the coset 119867 = 120587(1) hascompact neighborhood in 119866119867 Let 119870 ni 1 be a compact neighborhood in 119866 Choose a neighborhood119880 ni 1 such that 119880minus1113568119880 sub 119870 Claim 120587(119880) sub 120587(119870) Indeed if 119892119867 isin 120587(119880) then the neighborhood 119880119892119867intersects 120587(119880) that is 119906119892119867 = 119906prime119867 for some 119906 119906prime isin 119880 Hence 119892119867 = 119906minus1113568119906prime119867 isin 120587(119880minus1113568119880) sub 120587(119870)Therefore 120587(119880) ni 120587(1) is an open neighborhood with 120587(119880) compact since 120587(119870) is compact

If 119866119867 is Hausdorff (resp discrete) then 119867 = 120587minus1113568(120587(1)) is closed (resp open) Conversely supposethat 119867 is closed in 119866 Given cosets 119909119867 ne 119910119867 choose an open neighborhood 119881 ni 1 in 119866 such that119881119909cap 119910119867 = empty equivalently 119881119909119867 cap 119910119867 = empty Then choose an open 119880 ni 1 in 119866 such that 119880minus1113568119880 sub 119881 Itfollows that 119880119909119867 cap 119880119910119867 = empty and these are disjoint open neighborhoods of 120587(119909) and 120587(119910) All in all119866119867 is Hausdorff

On the other hand 119867 is open implies 120587(119867) is open thus all singletons in 119866119867 are open whence thediscreteness

As for 119867119866 we pass to 119866op

Lemma 15 Let119866 be a locally compact group (not necessarily Hausdorff) and let119867 sub 119866 be a subgroupEvery compact subset of 119867119866 is the image of some compact subset of 119866

Proof Let 119880 ni 1 be an open neighborhood in 119866 with compact closure 119880 For every compact subset 119870of 119867119866 we have an open covering 119870 sub ⋃119892isin119866 120587(119892119880) hence there exists 1198921113568 hellip 119892119899 isin 119866 such that

119870 sub1198991113996119894=1113568

120587(119892119894119880) = 120587

⎛⎜⎜⎜⎜⎜⎝1198991113996119894=1113568

119892119894119880

⎞⎟⎟⎟⎟⎟⎠ sub 120587

⎛⎜⎜⎜⎜⎜⎝1198991113996119894=1113568

119892119894

⎞⎟⎟⎟⎟⎟⎠

Take the compact subset 119870 ∶= ⋃119899119894=1113568 119892119894 cap 120587minus1113568(119870) and observe that 120587(119870) = 119870

Remark 16 The following operations on topological groups are evident

1 Let 119867 sub 119866 be a closed normal subgroup then 119866119867 is a topological group by Proposition 14locally compact if 119866 is We leave it to the reader to characterize 119866119867 by universal properties inTopGrp

3

2 If 1198661113568 1198661113569 are topological groups then 1198661113568 times 1198661113569 is naturally a topological group Again it hasan outright categorical characterization the product in TopGrp More generally one can formfibered products in TopGrp

3 In a similar vein limlarrminusminus exist in TopGrp their underlying abstract groups are just the usual limlarrminusminus

Harmonic analysis in its classical sense applies mainly to locally compact topological groups Here-after we adopt the shorthand locally compact groupsRemark 17 The family of locally compact groups is closed under passing to closed subgroups Hausdorffquotients and finite direct products However infinite direct products usually yield non-locally compactgroups This is one of the motivation for introducing restrict products into harmonic analysis

Example 18 Discrete groups are locally compact finite groups are compact

Example 19 The familiar groups (ℝ +) (ℂ +) are locally compact so are (ℂtimes sdot) and (ℝtimes sdot) Theidentity connected component ℝtimes

gt1113567 of ℝtimes is isomorphic to (ℝ +) through the logarithm The quotientgroup (ℝℤ +) ≃ 1201381113568 (via 119911 ↦ 1198901113569120587119894119911) is compact

12 Local fieldsJust as the case of groups a locally compact field is a field 119865 with a locally compact Hausdorff topologysuch that (119865 +) is a topological group and that (119909 119910) ↦ 119909119910 and 119909 ↦ 119909minus1113568 (on 119865times) are both continuous

Definition 110 A local field is a locally compact field that is not discrete

A detailed account of local fields can be found in any textbook on algebraic number theory Thetopology on a local field 119865 is always induced by an absolute value | sdot |119865 ∶ 119865 rarr ℝge1113567 satisfying

|119909|119865 = 0 ⟺ 119909 = 0 |1|119865 = 1|119909 + 119910|119865 le |119909|119865 + |119910|119865|119909119910|119865 = |119909| sdot |119910|119865

Furthermore 119865 is complete with respect to | sdot | Up to continuous isomorphisms local fields are classifiedas follows

Archimedean The fields ℝ and ℂ equipped with the usual absolute values

Non-archimedean characteristic zero The fieldsℚ119901 = ℤ119901[1113568119901 ] (the 119901-adic numbers where 119901 is a prime

number) or their finite extensions

Non-archimedean characteristic 119901 gt 0 The fields 120125119902((119905)) = 120125119902J119905K[ 1113568119905 ] of Laurent series in the variable119905 or their finite extensions where 119902 is a power of 119901 Here 120125119902 denotes the finite field of 119902 elements

Let 119865 be a non-archimedean local field It turns out the ultrametric inequality is satisfied

|119909 + 119910|119865 le max|119909|119865 |119910|119865 with equality when |119909| ne |119910|

Furthermore 120108119865 = 119909 ∶ |119909| le 1 is a subring called the ring of integers and 120109119865 = 119909 ∶ |119909| lt 1 is itsunique maximal ideal In fact 120109119865 is of the form (120603119865) here 120603119865 is called a uniformizer of 119865 and

119865times = 120603ℤ119865 times 120108times119865 120108times119865 = 119909 ∶ |119909| = 1

The normalized absolute value | sdot |119865 for non-archimedean 119865 is defined by

|120603119865 | = 119902minus1113568 119902 ∶= |120108119865120109119865 |

4

where 120603119865 is any uniformizer We will interpret | sdot |119865 in terms of modulus characters in 133If 119864 is a finite extension of any local field 119865 then 119864 is also local and | sdot |119865 admits a unique extension

to 119864 given by| sdot |119864 = |119873119864119865(sdot)|

1113568[119864∶119865]119865

where 119873119864119865 ∶ 119864 rarr 119865 is the norm map It defines the normalized absolute value on 119864 in the non-archimedean caseRemark 111 The example ℝ (resp ℚ119901) above is obtained by completing ℚ with respect to the usualabsolute value (resp the 119901-adic one |119909|119901 = 119901minus119907119901(119909) where 119907119901(119909) = 119896 if 119909 = 119901119896 119906119907 ne 0 with 119906 119907 isin ℤ coprimeto 119901 and 119907119901(0) = +infin)

Likewise 120125119902((119905)) is the completion of the function field120125119902(119905)with respect to |119909|1113567 = 119902minus1199071113577(119909) where 1199071113567(119909)is the vanishing order of the rational function 119909 at 0

In both cases the local field 119865 arises from completing some global fieldℚ or 120125119902(119905) or more generallytheir finite extensions The adjective ldquoglobalrdquo comes from geometry which is manifest in the case 120125119902(119905)it is the function field of the curveℙ1113568120125119902 and120125119902((119905)) should be thought as the function field on the ldquopuncturedformal diskrdquo at 0

13 Measures and integralsWe shall only consider Radon measures on a locally compact Hausdorff space 119883 These measures areby definition Borel locally finite and inner regular

Define the ℂ-vector space

119862119888(119883) ∶= 1114106119891 ∶ 119883 rarr ℂ continuous compactly supported1114109

= 1113996119870sub119883

compact

119862119888(119883 119870) 119862119888(119883 119870) ∶= 119891 isin 119862119888(119883) ∶ Supp(119891) sub 119870

We write 119862119888(119883)+ ∶= 1114106119891 isin 119862119888(119883) ∶ 119891 ge 01114109Remark 112 Following Bourbaki we identify positive Radon measures 120583 on 119883 and positive linearfunctionals 119868 = 119868120583 ∶ 119862119888(119883) rarr ℂ This means that 119868(119891) ge 0 if 119891 isin 119862119888(119883)+ In terms of integrals119868120583(119891) = int119883 119891 d120583 This is essentially a consequence of the Riesz representation theorem

Furthermore to prescribe a positive linear functional 119868 is the same as giving a function 119868 ∶ 119862119888(119883)+ rarrℝge1113567 that satisfies

bull 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)

bull 119868(119905119891) = 119905119868(119891) when 119905 isin ℝge1113567

To see this write 119891 = 119906 + 119894119907 isin 119862119888(119883) where 119906 119907 ∶ 119883 rarr ℝ furthermore any real-valued 119891 isin 119862119888(119883) canbe written as 119891 = 119891+ minus 119891minus as usual where 119891plusmn isin 119862119888(119883)+ All these decompositions are canonicalRemark 113 The complex Radon measures correspond to linear functionals 119868 ∶ 119862119888(119862) rarr ℂ such that119868 |119862119888(119883119870) is continuous with respect to sdot infin119870 ∶= sup119870 | sdot | for all 119870 In other words 119868 is continuous forthe topology of limminusminusrarr119870

119862119888(119883 119870) ≃ 119862119888(119883)

Now let119866 be a locally compact group and suppose that119883 is endowed with a continuous left119866-actionContinuity here means that the action map

119886 ∶ 119866 times 119883 ⟶ 119883(119892 119909)⟼ 119892119909

5

is continuous Similarly for right 119866-actions This action transports the functions 119891 isin 119862119888(119883) as well

119891119892 ∶= 1114094119909 ↦ 119891(119892119909)1114097 left action119892119891 ∶= 1114094119909 ↦ 119891(119909119892)]1114097 right action

These terminologies are justified as 119891119892119892prime = 11141001198911198921114103119892prime

and 119892119892prime119891 = 119892 1114100119892prime1198911114103 they also preserve 119862119888(119883)+ There-fore for 119866 acting on the left (resp on the right) of 119883 it also acts from the same side on the space ofpositive Radon measures on 119883 by transport of structure In terms of positive linear functionals

119868 ⟼ 1114094119892119868 ∶ 119891 ↦ 119868(119891119892)1114097 left action

119868 ⟼ 1114094119868119892 ∶ 119891 ↦ 119868(119892119891)1114097 right action

This pair of definitions is swapped under 119866 119866opBy taking transposes 119866 also transports complex Radon measures (Remark 113) Mnemonic tech-

niqued(119892120583)(119909) = d120583(119892minus1113568119909) d(120583119892)(119909) = d120583(119909119892minus1113568)

since a familiar change of variables yields

1114009119883119891(119909) d(119892120583)(119909) transpose= 119868(119891119892) = 1114009

119883119891(119892119909) d120583(119909) = 1114009

119883119891(119909) d120583(119892minus1113568119909)

1114009119883119891(119909) d(120583119892)(119909) transpose= 119868(119892119891) = 1114009

119883119891(119909119892) d120583(119909) = 1114009

119883119891(119909) d120583(119909119892minus1113568)

and 119891 isin 119862119888(119883) is arbitrary These formulas extend to all 119891 isin 1198711113568(119883 120583) by approximation

Definition 114 Suppose that 119866 acts continuous on the left (resp right) of 119883 Let 120594 ∶ 119866 rarr ℝtimesgt1113567

by a continuous homomorphism We say that a complex Radon measure 120583 is quasi-invariant or aneigenmeasure with eigencharacter 120594 if 119892120583 = 120594(119892)minus1113568120583 (resp 120583119892 = 120594(119892)minus1113568120583) for all 119892 isin 119866 This can alsobe expressed as

d120583(119892119909) = 120594(119892) d120583(119909) left actiond120583(119909119892) = 120594(119892) d120583(119909) right action

When 120594 = 1 we call 120583 an invariant measure

Observe that 120583 is quasi-invariant with eigencharacter 120594 under left119866-action if and only if it is so underthe right 119866op-actionRemark 115 Let 120594 120578 be continuous homomorphisms 119866 rarr ℂtimes Suppose that 119891 ∶ 119883 rarr ℂ is contin-uous with 119891(119892119909) = 120578(119892)119891(119909) (resp 119891(119909119892) = 120578(119892)119891(119909)) for all 119892 and 119909 Then 120583 is quasi-invariant witheigencharacter 120594 if and only if 119891120583 is quasi-invariant with eigencharacter 120578120594

In particular we may let 119866 act on 119883 = 119866 by left (resp right) translations Therefore it makes senseto talk about left and right invariant measures on 119866

14 Haar measuresIn what follows measures are always nontrivial positive Radon measures

Definition 116 Let 119866 be a locally compact group A left (resp right) invariant measure on 119866 is calleda left (resp right) Haar measure

6

The group ℝtimesgt1113567 acts on the set of left (resp right) Haar measures by rescaling For commutative

groups we make no distinction of left and right

Example 117 If 119866 is discrete the counting measure Count(119864) = |119864| is a left and right Haar measureWhen 119866 is finite it is customary to take the normalized version 120583 ∶= |119866|minus1113568Count

Definition 118 Let 119866 be a locally compact For 119891 isin 119862119888(119866) we write 119891 ∶ 119909 ↦ 119891(119909minus1113568) 119891 isin 119862119888(119866)For any Radon measure 120583 on 119866 let be the Radon measure with d(119909) = d120583(119909minus1113568) in terms of linearfunctionals 119868(119891) = 119868120583( 119891)

Lemma 119 In the situation above 120583 is a left (resp right) Haar measure if and only if is a right(resp left) Haar measure

Proof An instance of transport of structure because 119909 ↦ 119909minus1113568 is an isomorphism of locally compactgroups 119866 simminusrarr 119866op

Theorem 120 (A Weil) For every locally compact group 119866 there exists a left (resp right) Haar mea-sure on 119866 They are unique up to ℝtimes

gt1113567-action

Proof The following arguments are taken from Bourbaki [1 VII sect12] Upon replacing 119866 by 119866op itsuffices to consider the case of left Haar measures For the existence part we seek a left 119866-invariantpositive linear functional on 119862119888(119866)+ Write

119862119888(119866)lowast+ ∶= 119862119888(119866)+ ∖ 0

For any compact subset 119870 sub 119866 define 119862119888(119866 119870)+ and 119862119888(119866 119870)lowast+ by intersecting 119862119888(119866 119870) with 119862119888(119866)+and 119862119888(119866)lowast+ Observe that for all 119891 isin 119862119888(119866)+ and 119892 isin 119862119888(119866)lowast+ there exist 119899 ge 1 1198881113568 hellip 119888119899 isin ℝge1113567 and1199041113568 hellip 119904119899 isin 119866 such that

119891 le 11988811135681198921199041113578 + hellip119888119899119892119904119899 119892119904119894(119909) ∶= 119892(119904119894119909)Indeed there is an open subset 119880 sub 119866 such that inf119880 119892 gt 0 now cover Supp(119891) by finitely many1199041113568119880hellip 119904119899119880

Given 119891 119892 as before define (119891 ∶ 119892) to be the infimum of 1198881113568+hellip+119888119899 among all choices of (1198881113568 hellip 1199041113568 hellip)satisfying the bound above We contend that

(119891119904 ∶ 119892) = (119891 ∶ 119892) 119904 isin 119866 (11)(119905119891 ∶ 119892) = 119905(119891 ∶ 119892) 119905 isin ℝge1113567 (12)

(1198911113568 + 1198911113569 ∶ 119892) le (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) (13)

(119891 ∶ 119892) ge sup 119891sup 119892 (14)

(119891 ∶ ℎ) le (119891 ∶ 119892)(119892 ∶ ℎ) ℎ 119892 isin 119862119888(119866)lowast+ (15)

0 lt 1(1198911113567 ∶ 119891)

le (119891 ∶ 119892)(1198911113567 ∶ 119892)

le (119891 ∶ 1198911113567) 119891 1198911113567 119892 isin 119862119888(119866)lowast+ (16)

and that for all 1198911113568 1198911113569 ℎ isin 119862119888(119866)+ with ℎ|Supp(1198911113578+1198911113579) ge 1 and all 120598 gt 0 there is a compact neighborhood119870 ni 1 such that

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1198911113568 + 1198911113569 ∶ 119892) + 120598(ℎ ∶ 119892) 119892 isin 119862119888(119866 119870)lowast+ (17)

The properties (11) mdash (14) are straightforward For (15) note that

119891 le 1114012119894119888119894119892119904119894 119892 le 1114012

119895119889119895ℎ119905119895 ⟹ 119891 le1114012

119894119895119888119894119889119895ℎ119905119895119904119894

7

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

To find 119880 set 119881 ∶= 119866 ∖ 119909119861minus1113568 For every 119886 isin 119860 sub 119881 there exists an open subset 119880 prime119886 ni 1 with

119880 prime119886119886 sub 119881 Take an open 119880119886 ni 1 with 119880119886119880119886 sub 119880 prime

119886 Compactness furnishes a finite subset 1198601113567 sub 119860 suchthat 119860 sub ⋃119905isin1198601113577 119880119905119905 We claim that 119880 ∶= ⋂119905isin1198601113577 119880119905 satisfies the requirements Indeed 119880 ni 1 and each119886 isin 119860 belongs to 119880119905119905 for some 119905 isin 1198601113567 hence

119880119886 sub 119880119880119905119905 sub 119880119905119880119905119905 sub 119880 prime119905 119905 sub 119881

thus 119880119860 sub 119881

Recall that a topological space is locally compact if every point has a compact neighborhood

Proposition 14 Let 119867 be a subgroup of a topological group 119866 (not necessarily Hausdorff) Endow119866119867 with the quotient topology

1 The quotient map 120587 ∶ 119866 rarr 119866119867 is open and continuous

2 If 119866 is locally compact so is 119866119867

3 119866119867 is Hausdorff (resp discrete) if and only if 119867 is closed (resp open)

The same holds for 119867119866

Proof The quotient map is always continuous If119880 sub 119866 is open then 120587minus1113568(120587(119880)) = 119880119867 is open by 13hence 120587(119880) is open

Suppose 119866 is locally compact By homogeneity it suffices to argue that the coset 119867 = 120587(1) hascompact neighborhood in 119866119867 Let 119870 ni 1 be a compact neighborhood in 119866 Choose a neighborhood119880 ni 1 such that 119880minus1113568119880 sub 119870 Claim 120587(119880) sub 120587(119870) Indeed if 119892119867 isin 120587(119880) then the neighborhood 119880119892119867intersects 120587(119880) that is 119906119892119867 = 119906prime119867 for some 119906 119906prime isin 119880 Hence 119892119867 = 119906minus1113568119906prime119867 isin 120587(119880minus1113568119880) sub 120587(119870)Therefore 120587(119880) ni 120587(1) is an open neighborhood with 120587(119880) compact since 120587(119870) is compact

If 119866119867 is Hausdorff (resp discrete) then 119867 = 120587minus1113568(120587(1)) is closed (resp open) Conversely supposethat 119867 is closed in 119866 Given cosets 119909119867 ne 119910119867 choose an open neighborhood 119881 ni 1 in 119866 such that119881119909cap 119910119867 = empty equivalently 119881119909119867 cap 119910119867 = empty Then choose an open 119880 ni 1 in 119866 such that 119880minus1113568119880 sub 119881 Itfollows that 119880119909119867 cap 119880119910119867 = empty and these are disjoint open neighborhoods of 120587(119909) and 120587(119910) All in all119866119867 is Hausdorff

On the other hand 119867 is open implies 120587(119867) is open thus all singletons in 119866119867 are open whence thediscreteness

As for 119867119866 we pass to 119866op

Lemma 15 Let119866 be a locally compact group (not necessarily Hausdorff) and let119867 sub 119866 be a subgroupEvery compact subset of 119867119866 is the image of some compact subset of 119866

Proof Let 119880 ni 1 be an open neighborhood in 119866 with compact closure 119880 For every compact subset 119870of 119867119866 we have an open covering 119870 sub ⋃119892isin119866 120587(119892119880) hence there exists 1198921113568 hellip 119892119899 isin 119866 such that

119870 sub1198991113996119894=1113568

120587(119892119894119880) = 120587

⎛⎜⎜⎜⎜⎜⎝1198991113996119894=1113568

119892119894119880

⎞⎟⎟⎟⎟⎟⎠ sub 120587

⎛⎜⎜⎜⎜⎜⎝1198991113996119894=1113568

119892119894

⎞⎟⎟⎟⎟⎟⎠

Take the compact subset 119870 ∶= ⋃119899119894=1113568 119892119894 cap 120587minus1113568(119870) and observe that 120587(119870) = 119870

Remark 16 The following operations on topological groups are evident

1 Let 119867 sub 119866 be a closed normal subgroup then 119866119867 is a topological group by Proposition 14locally compact if 119866 is We leave it to the reader to characterize 119866119867 by universal properties inTopGrp

3

2 If 1198661113568 1198661113569 are topological groups then 1198661113568 times 1198661113569 is naturally a topological group Again it hasan outright categorical characterization the product in TopGrp More generally one can formfibered products in TopGrp

3 In a similar vein limlarrminusminus exist in TopGrp their underlying abstract groups are just the usual limlarrminusminus

Harmonic analysis in its classical sense applies mainly to locally compact topological groups Here-after we adopt the shorthand locally compact groupsRemark 17 The family of locally compact groups is closed under passing to closed subgroups Hausdorffquotients and finite direct products However infinite direct products usually yield non-locally compactgroups This is one of the motivation for introducing restrict products into harmonic analysis

Example 18 Discrete groups are locally compact finite groups are compact

Example 19 The familiar groups (ℝ +) (ℂ +) are locally compact so are (ℂtimes sdot) and (ℝtimes sdot) Theidentity connected component ℝtimes

gt1113567 of ℝtimes is isomorphic to (ℝ +) through the logarithm The quotientgroup (ℝℤ +) ≃ 1201381113568 (via 119911 ↦ 1198901113569120587119894119911) is compact

12 Local fieldsJust as the case of groups a locally compact field is a field 119865 with a locally compact Hausdorff topologysuch that (119865 +) is a topological group and that (119909 119910) ↦ 119909119910 and 119909 ↦ 119909minus1113568 (on 119865times) are both continuous

Definition 110 A local field is a locally compact field that is not discrete

A detailed account of local fields can be found in any textbook on algebraic number theory Thetopology on a local field 119865 is always induced by an absolute value | sdot |119865 ∶ 119865 rarr ℝge1113567 satisfying

|119909|119865 = 0 ⟺ 119909 = 0 |1|119865 = 1|119909 + 119910|119865 le |119909|119865 + |119910|119865|119909119910|119865 = |119909| sdot |119910|119865

Furthermore 119865 is complete with respect to | sdot | Up to continuous isomorphisms local fields are classifiedas follows

Archimedean The fields ℝ and ℂ equipped with the usual absolute values

Non-archimedean characteristic zero The fieldsℚ119901 = ℤ119901[1113568119901 ] (the 119901-adic numbers where 119901 is a prime

number) or their finite extensions

Non-archimedean characteristic 119901 gt 0 The fields 120125119902((119905)) = 120125119902J119905K[ 1113568119905 ] of Laurent series in the variable119905 or their finite extensions where 119902 is a power of 119901 Here 120125119902 denotes the finite field of 119902 elements

Let 119865 be a non-archimedean local field It turns out the ultrametric inequality is satisfied

|119909 + 119910|119865 le max|119909|119865 |119910|119865 with equality when |119909| ne |119910|

Furthermore 120108119865 = 119909 ∶ |119909| le 1 is a subring called the ring of integers and 120109119865 = 119909 ∶ |119909| lt 1 is itsunique maximal ideal In fact 120109119865 is of the form (120603119865) here 120603119865 is called a uniformizer of 119865 and

119865times = 120603ℤ119865 times 120108times119865 120108times119865 = 119909 ∶ |119909| = 1

The normalized absolute value | sdot |119865 for non-archimedean 119865 is defined by

|120603119865 | = 119902minus1113568 119902 ∶= |120108119865120109119865 |

4

where 120603119865 is any uniformizer We will interpret | sdot |119865 in terms of modulus characters in 133If 119864 is a finite extension of any local field 119865 then 119864 is also local and | sdot |119865 admits a unique extension

to 119864 given by| sdot |119864 = |119873119864119865(sdot)|

1113568[119864∶119865]119865

where 119873119864119865 ∶ 119864 rarr 119865 is the norm map It defines the normalized absolute value on 119864 in the non-archimedean caseRemark 111 The example ℝ (resp ℚ119901) above is obtained by completing ℚ with respect to the usualabsolute value (resp the 119901-adic one |119909|119901 = 119901minus119907119901(119909) where 119907119901(119909) = 119896 if 119909 = 119901119896 119906119907 ne 0 with 119906 119907 isin ℤ coprimeto 119901 and 119907119901(0) = +infin)

Likewise 120125119902((119905)) is the completion of the function field120125119902(119905)with respect to |119909|1113567 = 119902minus1199071113577(119909) where 1199071113567(119909)is the vanishing order of the rational function 119909 at 0

In both cases the local field 119865 arises from completing some global fieldℚ or 120125119902(119905) or more generallytheir finite extensions The adjective ldquoglobalrdquo comes from geometry which is manifest in the case 120125119902(119905)it is the function field of the curveℙ1113568120125119902 and120125119902((119905)) should be thought as the function field on the ldquopuncturedformal diskrdquo at 0

13 Measures and integralsWe shall only consider Radon measures on a locally compact Hausdorff space 119883 These measures areby definition Borel locally finite and inner regular

Define the ℂ-vector space

119862119888(119883) ∶= 1114106119891 ∶ 119883 rarr ℂ continuous compactly supported1114109

= 1113996119870sub119883

compact

119862119888(119883 119870) 119862119888(119883 119870) ∶= 119891 isin 119862119888(119883) ∶ Supp(119891) sub 119870

We write 119862119888(119883)+ ∶= 1114106119891 isin 119862119888(119883) ∶ 119891 ge 01114109Remark 112 Following Bourbaki we identify positive Radon measures 120583 on 119883 and positive linearfunctionals 119868 = 119868120583 ∶ 119862119888(119883) rarr ℂ This means that 119868(119891) ge 0 if 119891 isin 119862119888(119883)+ In terms of integrals119868120583(119891) = int119883 119891 d120583 This is essentially a consequence of the Riesz representation theorem

Furthermore to prescribe a positive linear functional 119868 is the same as giving a function 119868 ∶ 119862119888(119883)+ rarrℝge1113567 that satisfies

bull 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)

bull 119868(119905119891) = 119905119868(119891) when 119905 isin ℝge1113567

To see this write 119891 = 119906 + 119894119907 isin 119862119888(119883) where 119906 119907 ∶ 119883 rarr ℝ furthermore any real-valued 119891 isin 119862119888(119883) canbe written as 119891 = 119891+ minus 119891minus as usual where 119891plusmn isin 119862119888(119883)+ All these decompositions are canonicalRemark 113 The complex Radon measures correspond to linear functionals 119868 ∶ 119862119888(119862) rarr ℂ such that119868 |119862119888(119883119870) is continuous with respect to sdot infin119870 ∶= sup119870 | sdot | for all 119870 In other words 119868 is continuous forthe topology of limminusminusrarr119870

119862119888(119883 119870) ≃ 119862119888(119883)

Now let119866 be a locally compact group and suppose that119883 is endowed with a continuous left119866-actionContinuity here means that the action map

119886 ∶ 119866 times 119883 ⟶ 119883(119892 119909)⟼ 119892119909

5

is continuous Similarly for right 119866-actions This action transports the functions 119891 isin 119862119888(119883) as well

119891119892 ∶= 1114094119909 ↦ 119891(119892119909)1114097 left action119892119891 ∶= 1114094119909 ↦ 119891(119909119892)]1114097 right action

These terminologies are justified as 119891119892119892prime = 11141001198911198921114103119892prime

and 119892119892prime119891 = 119892 1114100119892prime1198911114103 they also preserve 119862119888(119883)+ There-fore for 119866 acting on the left (resp on the right) of 119883 it also acts from the same side on the space ofpositive Radon measures on 119883 by transport of structure In terms of positive linear functionals

119868 ⟼ 1114094119892119868 ∶ 119891 ↦ 119868(119891119892)1114097 left action

119868 ⟼ 1114094119868119892 ∶ 119891 ↦ 119868(119892119891)1114097 right action

This pair of definitions is swapped under 119866 119866opBy taking transposes 119866 also transports complex Radon measures (Remark 113) Mnemonic tech-

niqued(119892120583)(119909) = d120583(119892minus1113568119909) d(120583119892)(119909) = d120583(119909119892minus1113568)

since a familiar change of variables yields

1114009119883119891(119909) d(119892120583)(119909) transpose= 119868(119891119892) = 1114009

119883119891(119892119909) d120583(119909) = 1114009

119883119891(119909) d120583(119892minus1113568119909)

1114009119883119891(119909) d(120583119892)(119909) transpose= 119868(119892119891) = 1114009

119883119891(119909119892) d120583(119909) = 1114009

119883119891(119909) d120583(119909119892minus1113568)

and 119891 isin 119862119888(119883) is arbitrary These formulas extend to all 119891 isin 1198711113568(119883 120583) by approximation

Definition 114 Suppose that 119866 acts continuous on the left (resp right) of 119883 Let 120594 ∶ 119866 rarr ℝtimesgt1113567

by a continuous homomorphism We say that a complex Radon measure 120583 is quasi-invariant or aneigenmeasure with eigencharacter 120594 if 119892120583 = 120594(119892)minus1113568120583 (resp 120583119892 = 120594(119892)minus1113568120583) for all 119892 isin 119866 This can alsobe expressed as

d120583(119892119909) = 120594(119892) d120583(119909) left actiond120583(119909119892) = 120594(119892) d120583(119909) right action

When 120594 = 1 we call 120583 an invariant measure

Observe that 120583 is quasi-invariant with eigencharacter 120594 under left119866-action if and only if it is so underthe right 119866op-actionRemark 115 Let 120594 120578 be continuous homomorphisms 119866 rarr ℂtimes Suppose that 119891 ∶ 119883 rarr ℂ is contin-uous with 119891(119892119909) = 120578(119892)119891(119909) (resp 119891(119909119892) = 120578(119892)119891(119909)) for all 119892 and 119909 Then 120583 is quasi-invariant witheigencharacter 120594 if and only if 119891120583 is quasi-invariant with eigencharacter 120578120594

In particular we may let 119866 act on 119883 = 119866 by left (resp right) translations Therefore it makes senseto talk about left and right invariant measures on 119866

14 Haar measuresIn what follows measures are always nontrivial positive Radon measures

Definition 116 Let 119866 be a locally compact group A left (resp right) invariant measure on 119866 is calleda left (resp right) Haar measure

6

The group ℝtimesgt1113567 acts on the set of left (resp right) Haar measures by rescaling For commutative

groups we make no distinction of left and right

Example 117 If 119866 is discrete the counting measure Count(119864) = |119864| is a left and right Haar measureWhen 119866 is finite it is customary to take the normalized version 120583 ∶= |119866|minus1113568Count

Definition 118 Let 119866 be a locally compact For 119891 isin 119862119888(119866) we write 119891 ∶ 119909 ↦ 119891(119909minus1113568) 119891 isin 119862119888(119866)For any Radon measure 120583 on 119866 let be the Radon measure with d(119909) = d120583(119909minus1113568) in terms of linearfunctionals 119868(119891) = 119868120583( 119891)

Lemma 119 In the situation above 120583 is a left (resp right) Haar measure if and only if is a right(resp left) Haar measure

Proof An instance of transport of structure because 119909 ↦ 119909minus1113568 is an isomorphism of locally compactgroups 119866 simminusrarr 119866op

Theorem 120 (A Weil) For every locally compact group 119866 there exists a left (resp right) Haar mea-sure on 119866 They are unique up to ℝtimes

gt1113567-action

Proof The following arguments are taken from Bourbaki [1 VII sect12] Upon replacing 119866 by 119866op itsuffices to consider the case of left Haar measures For the existence part we seek a left 119866-invariantpositive linear functional on 119862119888(119866)+ Write

119862119888(119866)lowast+ ∶= 119862119888(119866)+ ∖ 0

For any compact subset 119870 sub 119866 define 119862119888(119866 119870)+ and 119862119888(119866 119870)lowast+ by intersecting 119862119888(119866 119870) with 119862119888(119866)+and 119862119888(119866)lowast+ Observe that for all 119891 isin 119862119888(119866)+ and 119892 isin 119862119888(119866)lowast+ there exist 119899 ge 1 1198881113568 hellip 119888119899 isin ℝge1113567 and1199041113568 hellip 119904119899 isin 119866 such that

119891 le 11988811135681198921199041113578 + hellip119888119899119892119904119899 119892119904119894(119909) ∶= 119892(119904119894119909)Indeed there is an open subset 119880 sub 119866 such that inf119880 119892 gt 0 now cover Supp(119891) by finitely many1199041113568119880hellip 119904119899119880

Given 119891 119892 as before define (119891 ∶ 119892) to be the infimum of 1198881113568+hellip+119888119899 among all choices of (1198881113568 hellip 1199041113568 hellip)satisfying the bound above We contend that

(119891119904 ∶ 119892) = (119891 ∶ 119892) 119904 isin 119866 (11)(119905119891 ∶ 119892) = 119905(119891 ∶ 119892) 119905 isin ℝge1113567 (12)

(1198911113568 + 1198911113569 ∶ 119892) le (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) (13)

(119891 ∶ 119892) ge sup 119891sup 119892 (14)

(119891 ∶ ℎ) le (119891 ∶ 119892)(119892 ∶ ℎ) ℎ 119892 isin 119862119888(119866)lowast+ (15)

0 lt 1(1198911113567 ∶ 119891)

le (119891 ∶ 119892)(1198911113567 ∶ 119892)

le (119891 ∶ 1198911113567) 119891 1198911113567 119892 isin 119862119888(119866)lowast+ (16)

and that for all 1198911113568 1198911113569 ℎ isin 119862119888(119866)+ with ℎ|Supp(1198911113578+1198911113579) ge 1 and all 120598 gt 0 there is a compact neighborhood119870 ni 1 such that

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1198911113568 + 1198911113569 ∶ 119892) + 120598(ℎ ∶ 119892) 119892 isin 119862119888(119866 119870)lowast+ (17)

The properties (11) mdash (14) are straightforward For (15) note that

119891 le 1114012119894119888119894119892119904119894 119892 le 1114012

119895119889119895ℎ119905119895 ⟹ 119891 le1114012

119894119895119888119894119889119895ℎ119905119895119904119894

7

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

2 If 1198661113568 1198661113569 are topological groups then 1198661113568 times 1198661113569 is naturally a topological group Again it hasan outright categorical characterization the product in TopGrp More generally one can formfibered products in TopGrp

3 In a similar vein limlarrminusminus exist in TopGrp their underlying abstract groups are just the usual limlarrminusminus

Harmonic analysis in its classical sense applies mainly to locally compact topological groups Here-after we adopt the shorthand locally compact groupsRemark 17 The family of locally compact groups is closed under passing to closed subgroups Hausdorffquotients and finite direct products However infinite direct products usually yield non-locally compactgroups This is one of the motivation for introducing restrict products into harmonic analysis

Example 18 Discrete groups are locally compact finite groups are compact

Example 19 The familiar groups (ℝ +) (ℂ +) are locally compact so are (ℂtimes sdot) and (ℝtimes sdot) Theidentity connected component ℝtimes

gt1113567 of ℝtimes is isomorphic to (ℝ +) through the logarithm The quotientgroup (ℝℤ +) ≃ 1201381113568 (via 119911 ↦ 1198901113569120587119894119911) is compact

12 Local fieldsJust as the case of groups a locally compact field is a field 119865 with a locally compact Hausdorff topologysuch that (119865 +) is a topological group and that (119909 119910) ↦ 119909119910 and 119909 ↦ 119909minus1113568 (on 119865times) are both continuous

Definition 110 A local field is a locally compact field that is not discrete

A detailed account of local fields can be found in any textbook on algebraic number theory Thetopology on a local field 119865 is always induced by an absolute value | sdot |119865 ∶ 119865 rarr ℝge1113567 satisfying

|119909|119865 = 0 ⟺ 119909 = 0 |1|119865 = 1|119909 + 119910|119865 le |119909|119865 + |119910|119865|119909119910|119865 = |119909| sdot |119910|119865

Furthermore 119865 is complete with respect to | sdot | Up to continuous isomorphisms local fields are classifiedas follows

Archimedean The fields ℝ and ℂ equipped with the usual absolute values

Non-archimedean characteristic zero The fieldsℚ119901 = ℤ119901[1113568119901 ] (the 119901-adic numbers where 119901 is a prime

number) or their finite extensions

Non-archimedean characteristic 119901 gt 0 The fields 120125119902((119905)) = 120125119902J119905K[ 1113568119905 ] of Laurent series in the variable119905 or their finite extensions where 119902 is a power of 119901 Here 120125119902 denotes the finite field of 119902 elements

Let 119865 be a non-archimedean local field It turns out the ultrametric inequality is satisfied

|119909 + 119910|119865 le max|119909|119865 |119910|119865 with equality when |119909| ne |119910|

Furthermore 120108119865 = 119909 ∶ |119909| le 1 is a subring called the ring of integers and 120109119865 = 119909 ∶ |119909| lt 1 is itsunique maximal ideal In fact 120109119865 is of the form (120603119865) here 120603119865 is called a uniformizer of 119865 and

119865times = 120603ℤ119865 times 120108times119865 120108times119865 = 119909 ∶ |119909| = 1

The normalized absolute value | sdot |119865 for non-archimedean 119865 is defined by

|120603119865 | = 119902minus1113568 119902 ∶= |120108119865120109119865 |

4

where 120603119865 is any uniformizer We will interpret | sdot |119865 in terms of modulus characters in 133If 119864 is a finite extension of any local field 119865 then 119864 is also local and | sdot |119865 admits a unique extension

to 119864 given by| sdot |119864 = |119873119864119865(sdot)|

1113568[119864∶119865]119865

where 119873119864119865 ∶ 119864 rarr 119865 is the norm map It defines the normalized absolute value on 119864 in the non-archimedean caseRemark 111 The example ℝ (resp ℚ119901) above is obtained by completing ℚ with respect to the usualabsolute value (resp the 119901-adic one |119909|119901 = 119901minus119907119901(119909) where 119907119901(119909) = 119896 if 119909 = 119901119896 119906119907 ne 0 with 119906 119907 isin ℤ coprimeto 119901 and 119907119901(0) = +infin)

Likewise 120125119902((119905)) is the completion of the function field120125119902(119905)with respect to |119909|1113567 = 119902minus1199071113577(119909) where 1199071113567(119909)is the vanishing order of the rational function 119909 at 0

In both cases the local field 119865 arises from completing some global fieldℚ or 120125119902(119905) or more generallytheir finite extensions The adjective ldquoglobalrdquo comes from geometry which is manifest in the case 120125119902(119905)it is the function field of the curveℙ1113568120125119902 and120125119902((119905)) should be thought as the function field on the ldquopuncturedformal diskrdquo at 0

13 Measures and integralsWe shall only consider Radon measures on a locally compact Hausdorff space 119883 These measures areby definition Borel locally finite and inner regular

Define the ℂ-vector space

119862119888(119883) ∶= 1114106119891 ∶ 119883 rarr ℂ continuous compactly supported1114109

= 1113996119870sub119883

compact

119862119888(119883 119870) 119862119888(119883 119870) ∶= 119891 isin 119862119888(119883) ∶ Supp(119891) sub 119870

We write 119862119888(119883)+ ∶= 1114106119891 isin 119862119888(119883) ∶ 119891 ge 01114109Remark 112 Following Bourbaki we identify positive Radon measures 120583 on 119883 and positive linearfunctionals 119868 = 119868120583 ∶ 119862119888(119883) rarr ℂ This means that 119868(119891) ge 0 if 119891 isin 119862119888(119883)+ In terms of integrals119868120583(119891) = int119883 119891 d120583 This is essentially a consequence of the Riesz representation theorem

Furthermore to prescribe a positive linear functional 119868 is the same as giving a function 119868 ∶ 119862119888(119883)+ rarrℝge1113567 that satisfies

bull 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)

bull 119868(119905119891) = 119905119868(119891) when 119905 isin ℝge1113567

To see this write 119891 = 119906 + 119894119907 isin 119862119888(119883) where 119906 119907 ∶ 119883 rarr ℝ furthermore any real-valued 119891 isin 119862119888(119883) canbe written as 119891 = 119891+ minus 119891minus as usual where 119891plusmn isin 119862119888(119883)+ All these decompositions are canonicalRemark 113 The complex Radon measures correspond to linear functionals 119868 ∶ 119862119888(119862) rarr ℂ such that119868 |119862119888(119883119870) is continuous with respect to sdot infin119870 ∶= sup119870 | sdot | for all 119870 In other words 119868 is continuous forthe topology of limminusminusrarr119870

119862119888(119883 119870) ≃ 119862119888(119883)

Now let119866 be a locally compact group and suppose that119883 is endowed with a continuous left119866-actionContinuity here means that the action map

119886 ∶ 119866 times 119883 ⟶ 119883(119892 119909)⟼ 119892119909

5

is continuous Similarly for right 119866-actions This action transports the functions 119891 isin 119862119888(119883) as well

119891119892 ∶= 1114094119909 ↦ 119891(119892119909)1114097 left action119892119891 ∶= 1114094119909 ↦ 119891(119909119892)]1114097 right action

These terminologies are justified as 119891119892119892prime = 11141001198911198921114103119892prime

and 119892119892prime119891 = 119892 1114100119892prime1198911114103 they also preserve 119862119888(119883)+ There-fore for 119866 acting on the left (resp on the right) of 119883 it also acts from the same side on the space ofpositive Radon measures on 119883 by transport of structure In terms of positive linear functionals

119868 ⟼ 1114094119892119868 ∶ 119891 ↦ 119868(119891119892)1114097 left action

119868 ⟼ 1114094119868119892 ∶ 119891 ↦ 119868(119892119891)1114097 right action

This pair of definitions is swapped under 119866 119866opBy taking transposes 119866 also transports complex Radon measures (Remark 113) Mnemonic tech-

niqued(119892120583)(119909) = d120583(119892minus1113568119909) d(120583119892)(119909) = d120583(119909119892minus1113568)

since a familiar change of variables yields

1114009119883119891(119909) d(119892120583)(119909) transpose= 119868(119891119892) = 1114009

119883119891(119892119909) d120583(119909) = 1114009

119883119891(119909) d120583(119892minus1113568119909)

1114009119883119891(119909) d(120583119892)(119909) transpose= 119868(119892119891) = 1114009

119883119891(119909119892) d120583(119909) = 1114009

119883119891(119909) d120583(119909119892minus1113568)

and 119891 isin 119862119888(119883) is arbitrary These formulas extend to all 119891 isin 1198711113568(119883 120583) by approximation

Definition 114 Suppose that 119866 acts continuous on the left (resp right) of 119883 Let 120594 ∶ 119866 rarr ℝtimesgt1113567

by a continuous homomorphism We say that a complex Radon measure 120583 is quasi-invariant or aneigenmeasure with eigencharacter 120594 if 119892120583 = 120594(119892)minus1113568120583 (resp 120583119892 = 120594(119892)minus1113568120583) for all 119892 isin 119866 This can alsobe expressed as

d120583(119892119909) = 120594(119892) d120583(119909) left actiond120583(119909119892) = 120594(119892) d120583(119909) right action

When 120594 = 1 we call 120583 an invariant measure

Observe that 120583 is quasi-invariant with eigencharacter 120594 under left119866-action if and only if it is so underthe right 119866op-actionRemark 115 Let 120594 120578 be continuous homomorphisms 119866 rarr ℂtimes Suppose that 119891 ∶ 119883 rarr ℂ is contin-uous with 119891(119892119909) = 120578(119892)119891(119909) (resp 119891(119909119892) = 120578(119892)119891(119909)) for all 119892 and 119909 Then 120583 is quasi-invariant witheigencharacter 120594 if and only if 119891120583 is quasi-invariant with eigencharacter 120578120594

In particular we may let 119866 act on 119883 = 119866 by left (resp right) translations Therefore it makes senseto talk about left and right invariant measures on 119866

14 Haar measuresIn what follows measures are always nontrivial positive Radon measures

Definition 116 Let 119866 be a locally compact group A left (resp right) invariant measure on 119866 is calleda left (resp right) Haar measure

6

The group ℝtimesgt1113567 acts on the set of left (resp right) Haar measures by rescaling For commutative

groups we make no distinction of left and right

Example 117 If 119866 is discrete the counting measure Count(119864) = |119864| is a left and right Haar measureWhen 119866 is finite it is customary to take the normalized version 120583 ∶= |119866|minus1113568Count

Definition 118 Let 119866 be a locally compact For 119891 isin 119862119888(119866) we write 119891 ∶ 119909 ↦ 119891(119909minus1113568) 119891 isin 119862119888(119866)For any Radon measure 120583 on 119866 let be the Radon measure with d(119909) = d120583(119909minus1113568) in terms of linearfunctionals 119868(119891) = 119868120583( 119891)

Lemma 119 In the situation above 120583 is a left (resp right) Haar measure if and only if is a right(resp left) Haar measure

Proof An instance of transport of structure because 119909 ↦ 119909minus1113568 is an isomorphism of locally compactgroups 119866 simminusrarr 119866op

Theorem 120 (A Weil) For every locally compact group 119866 there exists a left (resp right) Haar mea-sure on 119866 They are unique up to ℝtimes

gt1113567-action

Proof The following arguments are taken from Bourbaki [1 VII sect12] Upon replacing 119866 by 119866op itsuffices to consider the case of left Haar measures For the existence part we seek a left 119866-invariantpositive linear functional on 119862119888(119866)+ Write

119862119888(119866)lowast+ ∶= 119862119888(119866)+ ∖ 0

For any compact subset 119870 sub 119866 define 119862119888(119866 119870)+ and 119862119888(119866 119870)lowast+ by intersecting 119862119888(119866 119870) with 119862119888(119866)+and 119862119888(119866)lowast+ Observe that for all 119891 isin 119862119888(119866)+ and 119892 isin 119862119888(119866)lowast+ there exist 119899 ge 1 1198881113568 hellip 119888119899 isin ℝge1113567 and1199041113568 hellip 119904119899 isin 119866 such that

119891 le 11988811135681198921199041113578 + hellip119888119899119892119904119899 119892119904119894(119909) ∶= 119892(119904119894119909)Indeed there is an open subset 119880 sub 119866 such that inf119880 119892 gt 0 now cover Supp(119891) by finitely many1199041113568119880hellip 119904119899119880

Given 119891 119892 as before define (119891 ∶ 119892) to be the infimum of 1198881113568+hellip+119888119899 among all choices of (1198881113568 hellip 1199041113568 hellip)satisfying the bound above We contend that

(119891119904 ∶ 119892) = (119891 ∶ 119892) 119904 isin 119866 (11)(119905119891 ∶ 119892) = 119905(119891 ∶ 119892) 119905 isin ℝge1113567 (12)

(1198911113568 + 1198911113569 ∶ 119892) le (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) (13)

(119891 ∶ 119892) ge sup 119891sup 119892 (14)

(119891 ∶ ℎ) le (119891 ∶ 119892)(119892 ∶ ℎ) ℎ 119892 isin 119862119888(119866)lowast+ (15)

0 lt 1(1198911113567 ∶ 119891)

le (119891 ∶ 119892)(1198911113567 ∶ 119892)

le (119891 ∶ 1198911113567) 119891 1198911113567 119892 isin 119862119888(119866)lowast+ (16)

and that for all 1198911113568 1198911113569 ℎ isin 119862119888(119866)+ with ℎ|Supp(1198911113578+1198911113579) ge 1 and all 120598 gt 0 there is a compact neighborhood119870 ni 1 such that

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1198911113568 + 1198911113569 ∶ 119892) + 120598(ℎ ∶ 119892) 119892 isin 119862119888(119866 119870)lowast+ (17)

The properties (11) mdash (14) are straightforward For (15) note that

119891 le 1114012119894119888119894119892119904119894 119892 le 1114012

119895119889119895ℎ119905119895 ⟹ 119891 le1114012

119894119895119888119894119889119895ℎ119905119895119904119894

7

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

where 120603119865 is any uniformizer We will interpret | sdot |119865 in terms of modulus characters in 133If 119864 is a finite extension of any local field 119865 then 119864 is also local and | sdot |119865 admits a unique extension

to 119864 given by| sdot |119864 = |119873119864119865(sdot)|

1113568[119864∶119865]119865

where 119873119864119865 ∶ 119864 rarr 119865 is the norm map It defines the normalized absolute value on 119864 in the non-archimedean caseRemark 111 The example ℝ (resp ℚ119901) above is obtained by completing ℚ with respect to the usualabsolute value (resp the 119901-adic one |119909|119901 = 119901minus119907119901(119909) where 119907119901(119909) = 119896 if 119909 = 119901119896 119906119907 ne 0 with 119906 119907 isin ℤ coprimeto 119901 and 119907119901(0) = +infin)

Likewise 120125119902((119905)) is the completion of the function field120125119902(119905)with respect to |119909|1113567 = 119902minus1199071113577(119909) where 1199071113567(119909)is the vanishing order of the rational function 119909 at 0

In both cases the local field 119865 arises from completing some global fieldℚ or 120125119902(119905) or more generallytheir finite extensions The adjective ldquoglobalrdquo comes from geometry which is manifest in the case 120125119902(119905)it is the function field of the curveℙ1113568120125119902 and120125119902((119905)) should be thought as the function field on the ldquopuncturedformal diskrdquo at 0

13 Measures and integralsWe shall only consider Radon measures on a locally compact Hausdorff space 119883 These measures areby definition Borel locally finite and inner regular

Define the ℂ-vector space

119862119888(119883) ∶= 1114106119891 ∶ 119883 rarr ℂ continuous compactly supported1114109

= 1113996119870sub119883

compact

119862119888(119883 119870) 119862119888(119883 119870) ∶= 119891 isin 119862119888(119883) ∶ Supp(119891) sub 119870

We write 119862119888(119883)+ ∶= 1114106119891 isin 119862119888(119883) ∶ 119891 ge 01114109Remark 112 Following Bourbaki we identify positive Radon measures 120583 on 119883 and positive linearfunctionals 119868 = 119868120583 ∶ 119862119888(119883) rarr ℂ This means that 119868(119891) ge 0 if 119891 isin 119862119888(119883)+ In terms of integrals119868120583(119891) = int119883 119891 d120583 This is essentially a consequence of the Riesz representation theorem

Furthermore to prescribe a positive linear functional 119868 is the same as giving a function 119868 ∶ 119862119888(119883)+ rarrℝge1113567 that satisfies

bull 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)

bull 119868(119905119891) = 119905119868(119891) when 119905 isin ℝge1113567

To see this write 119891 = 119906 + 119894119907 isin 119862119888(119883) where 119906 119907 ∶ 119883 rarr ℝ furthermore any real-valued 119891 isin 119862119888(119883) canbe written as 119891 = 119891+ minus 119891minus as usual where 119891plusmn isin 119862119888(119883)+ All these decompositions are canonicalRemark 113 The complex Radon measures correspond to linear functionals 119868 ∶ 119862119888(119862) rarr ℂ such that119868 |119862119888(119883119870) is continuous with respect to sdot infin119870 ∶= sup119870 | sdot | for all 119870 In other words 119868 is continuous forthe topology of limminusminusrarr119870

119862119888(119883 119870) ≃ 119862119888(119883)

Now let119866 be a locally compact group and suppose that119883 is endowed with a continuous left119866-actionContinuity here means that the action map

119886 ∶ 119866 times 119883 ⟶ 119883(119892 119909)⟼ 119892119909

5

is continuous Similarly for right 119866-actions This action transports the functions 119891 isin 119862119888(119883) as well

119891119892 ∶= 1114094119909 ↦ 119891(119892119909)1114097 left action119892119891 ∶= 1114094119909 ↦ 119891(119909119892)]1114097 right action

These terminologies are justified as 119891119892119892prime = 11141001198911198921114103119892prime

and 119892119892prime119891 = 119892 1114100119892prime1198911114103 they also preserve 119862119888(119883)+ There-fore for 119866 acting on the left (resp on the right) of 119883 it also acts from the same side on the space ofpositive Radon measures on 119883 by transport of structure In terms of positive linear functionals

119868 ⟼ 1114094119892119868 ∶ 119891 ↦ 119868(119891119892)1114097 left action

119868 ⟼ 1114094119868119892 ∶ 119891 ↦ 119868(119892119891)1114097 right action

This pair of definitions is swapped under 119866 119866opBy taking transposes 119866 also transports complex Radon measures (Remark 113) Mnemonic tech-

niqued(119892120583)(119909) = d120583(119892minus1113568119909) d(120583119892)(119909) = d120583(119909119892minus1113568)

since a familiar change of variables yields

1114009119883119891(119909) d(119892120583)(119909) transpose= 119868(119891119892) = 1114009

119883119891(119892119909) d120583(119909) = 1114009

119883119891(119909) d120583(119892minus1113568119909)

1114009119883119891(119909) d(120583119892)(119909) transpose= 119868(119892119891) = 1114009

119883119891(119909119892) d120583(119909) = 1114009

119883119891(119909) d120583(119909119892minus1113568)

and 119891 isin 119862119888(119883) is arbitrary These formulas extend to all 119891 isin 1198711113568(119883 120583) by approximation

Definition 114 Suppose that 119866 acts continuous on the left (resp right) of 119883 Let 120594 ∶ 119866 rarr ℝtimesgt1113567

by a continuous homomorphism We say that a complex Radon measure 120583 is quasi-invariant or aneigenmeasure with eigencharacter 120594 if 119892120583 = 120594(119892)minus1113568120583 (resp 120583119892 = 120594(119892)minus1113568120583) for all 119892 isin 119866 This can alsobe expressed as

d120583(119892119909) = 120594(119892) d120583(119909) left actiond120583(119909119892) = 120594(119892) d120583(119909) right action

When 120594 = 1 we call 120583 an invariant measure

Observe that 120583 is quasi-invariant with eigencharacter 120594 under left119866-action if and only if it is so underthe right 119866op-actionRemark 115 Let 120594 120578 be continuous homomorphisms 119866 rarr ℂtimes Suppose that 119891 ∶ 119883 rarr ℂ is contin-uous with 119891(119892119909) = 120578(119892)119891(119909) (resp 119891(119909119892) = 120578(119892)119891(119909)) for all 119892 and 119909 Then 120583 is quasi-invariant witheigencharacter 120594 if and only if 119891120583 is quasi-invariant with eigencharacter 120578120594

In particular we may let 119866 act on 119883 = 119866 by left (resp right) translations Therefore it makes senseto talk about left and right invariant measures on 119866

14 Haar measuresIn what follows measures are always nontrivial positive Radon measures

Definition 116 Let 119866 be a locally compact group A left (resp right) invariant measure on 119866 is calleda left (resp right) Haar measure

6

The group ℝtimesgt1113567 acts on the set of left (resp right) Haar measures by rescaling For commutative

groups we make no distinction of left and right

Example 117 If 119866 is discrete the counting measure Count(119864) = |119864| is a left and right Haar measureWhen 119866 is finite it is customary to take the normalized version 120583 ∶= |119866|minus1113568Count

Definition 118 Let 119866 be a locally compact For 119891 isin 119862119888(119866) we write 119891 ∶ 119909 ↦ 119891(119909minus1113568) 119891 isin 119862119888(119866)For any Radon measure 120583 on 119866 let be the Radon measure with d(119909) = d120583(119909minus1113568) in terms of linearfunctionals 119868(119891) = 119868120583( 119891)

Lemma 119 In the situation above 120583 is a left (resp right) Haar measure if and only if is a right(resp left) Haar measure

Proof An instance of transport of structure because 119909 ↦ 119909minus1113568 is an isomorphism of locally compactgroups 119866 simminusrarr 119866op

Theorem 120 (A Weil) For every locally compact group 119866 there exists a left (resp right) Haar mea-sure on 119866 They are unique up to ℝtimes

gt1113567-action

Proof The following arguments are taken from Bourbaki [1 VII sect12] Upon replacing 119866 by 119866op itsuffices to consider the case of left Haar measures For the existence part we seek a left 119866-invariantpositive linear functional on 119862119888(119866)+ Write

119862119888(119866)lowast+ ∶= 119862119888(119866)+ ∖ 0

For any compact subset 119870 sub 119866 define 119862119888(119866 119870)+ and 119862119888(119866 119870)lowast+ by intersecting 119862119888(119866 119870) with 119862119888(119866)+and 119862119888(119866)lowast+ Observe that for all 119891 isin 119862119888(119866)+ and 119892 isin 119862119888(119866)lowast+ there exist 119899 ge 1 1198881113568 hellip 119888119899 isin ℝge1113567 and1199041113568 hellip 119904119899 isin 119866 such that

119891 le 11988811135681198921199041113578 + hellip119888119899119892119904119899 119892119904119894(119909) ∶= 119892(119904119894119909)Indeed there is an open subset 119880 sub 119866 such that inf119880 119892 gt 0 now cover Supp(119891) by finitely many1199041113568119880hellip 119904119899119880

Given 119891 119892 as before define (119891 ∶ 119892) to be the infimum of 1198881113568+hellip+119888119899 among all choices of (1198881113568 hellip 1199041113568 hellip)satisfying the bound above We contend that

(119891119904 ∶ 119892) = (119891 ∶ 119892) 119904 isin 119866 (11)(119905119891 ∶ 119892) = 119905(119891 ∶ 119892) 119905 isin ℝge1113567 (12)

(1198911113568 + 1198911113569 ∶ 119892) le (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) (13)

(119891 ∶ 119892) ge sup 119891sup 119892 (14)

(119891 ∶ ℎ) le (119891 ∶ 119892)(119892 ∶ ℎ) ℎ 119892 isin 119862119888(119866)lowast+ (15)

0 lt 1(1198911113567 ∶ 119891)

le (119891 ∶ 119892)(1198911113567 ∶ 119892)

le (119891 ∶ 1198911113567) 119891 1198911113567 119892 isin 119862119888(119866)lowast+ (16)

and that for all 1198911113568 1198911113569 ℎ isin 119862119888(119866)+ with ℎ|Supp(1198911113578+1198911113579) ge 1 and all 120598 gt 0 there is a compact neighborhood119870 ni 1 such that

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1198911113568 + 1198911113569 ∶ 119892) + 120598(ℎ ∶ 119892) 119892 isin 119862119888(119866 119870)lowast+ (17)

The properties (11) mdash (14) are straightforward For (15) note that

119891 le 1114012119894119888119894119892119904119894 119892 le 1114012

119895119889119895ℎ119905119895 ⟹ 119891 le1114012

119894119895119888119894119889119895ℎ119905119895119904119894

7

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

is continuous Similarly for right 119866-actions This action transports the functions 119891 isin 119862119888(119883) as well

119891119892 ∶= 1114094119909 ↦ 119891(119892119909)1114097 left action119892119891 ∶= 1114094119909 ↦ 119891(119909119892)]1114097 right action

These terminologies are justified as 119891119892119892prime = 11141001198911198921114103119892prime

and 119892119892prime119891 = 119892 1114100119892prime1198911114103 they also preserve 119862119888(119883)+ There-fore for 119866 acting on the left (resp on the right) of 119883 it also acts from the same side on the space ofpositive Radon measures on 119883 by transport of structure In terms of positive linear functionals

119868 ⟼ 1114094119892119868 ∶ 119891 ↦ 119868(119891119892)1114097 left action

119868 ⟼ 1114094119868119892 ∶ 119891 ↦ 119868(119892119891)1114097 right action

This pair of definitions is swapped under 119866 119866opBy taking transposes 119866 also transports complex Radon measures (Remark 113) Mnemonic tech-

niqued(119892120583)(119909) = d120583(119892minus1113568119909) d(120583119892)(119909) = d120583(119909119892minus1113568)

since a familiar change of variables yields

1114009119883119891(119909) d(119892120583)(119909) transpose= 119868(119891119892) = 1114009

119883119891(119892119909) d120583(119909) = 1114009

119883119891(119909) d120583(119892minus1113568119909)

1114009119883119891(119909) d(120583119892)(119909) transpose= 119868(119892119891) = 1114009

119883119891(119909119892) d120583(119909) = 1114009

119883119891(119909) d120583(119909119892minus1113568)

and 119891 isin 119862119888(119883) is arbitrary These formulas extend to all 119891 isin 1198711113568(119883 120583) by approximation

Definition 114 Suppose that 119866 acts continuous on the left (resp right) of 119883 Let 120594 ∶ 119866 rarr ℝtimesgt1113567

by a continuous homomorphism We say that a complex Radon measure 120583 is quasi-invariant or aneigenmeasure with eigencharacter 120594 if 119892120583 = 120594(119892)minus1113568120583 (resp 120583119892 = 120594(119892)minus1113568120583) for all 119892 isin 119866 This can alsobe expressed as

d120583(119892119909) = 120594(119892) d120583(119909) left actiond120583(119909119892) = 120594(119892) d120583(119909) right action

When 120594 = 1 we call 120583 an invariant measure

Observe that 120583 is quasi-invariant with eigencharacter 120594 under left119866-action if and only if it is so underthe right 119866op-actionRemark 115 Let 120594 120578 be continuous homomorphisms 119866 rarr ℂtimes Suppose that 119891 ∶ 119883 rarr ℂ is contin-uous with 119891(119892119909) = 120578(119892)119891(119909) (resp 119891(119909119892) = 120578(119892)119891(119909)) for all 119892 and 119909 Then 120583 is quasi-invariant witheigencharacter 120594 if and only if 119891120583 is quasi-invariant with eigencharacter 120578120594

In particular we may let 119866 act on 119883 = 119866 by left (resp right) translations Therefore it makes senseto talk about left and right invariant measures on 119866

14 Haar measuresIn what follows measures are always nontrivial positive Radon measures

Definition 116 Let 119866 be a locally compact group A left (resp right) invariant measure on 119866 is calleda left (resp right) Haar measure

6

The group ℝtimesgt1113567 acts on the set of left (resp right) Haar measures by rescaling For commutative

groups we make no distinction of left and right

Example 117 If 119866 is discrete the counting measure Count(119864) = |119864| is a left and right Haar measureWhen 119866 is finite it is customary to take the normalized version 120583 ∶= |119866|minus1113568Count

Definition 118 Let 119866 be a locally compact For 119891 isin 119862119888(119866) we write 119891 ∶ 119909 ↦ 119891(119909minus1113568) 119891 isin 119862119888(119866)For any Radon measure 120583 on 119866 let be the Radon measure with d(119909) = d120583(119909minus1113568) in terms of linearfunctionals 119868(119891) = 119868120583( 119891)

Lemma 119 In the situation above 120583 is a left (resp right) Haar measure if and only if is a right(resp left) Haar measure

Proof An instance of transport of structure because 119909 ↦ 119909minus1113568 is an isomorphism of locally compactgroups 119866 simminusrarr 119866op

Theorem 120 (A Weil) For every locally compact group 119866 there exists a left (resp right) Haar mea-sure on 119866 They are unique up to ℝtimes

gt1113567-action

Proof The following arguments are taken from Bourbaki [1 VII sect12] Upon replacing 119866 by 119866op itsuffices to consider the case of left Haar measures For the existence part we seek a left 119866-invariantpositive linear functional on 119862119888(119866)+ Write

119862119888(119866)lowast+ ∶= 119862119888(119866)+ ∖ 0

For any compact subset 119870 sub 119866 define 119862119888(119866 119870)+ and 119862119888(119866 119870)lowast+ by intersecting 119862119888(119866 119870) with 119862119888(119866)+and 119862119888(119866)lowast+ Observe that for all 119891 isin 119862119888(119866)+ and 119892 isin 119862119888(119866)lowast+ there exist 119899 ge 1 1198881113568 hellip 119888119899 isin ℝge1113567 and1199041113568 hellip 119904119899 isin 119866 such that

119891 le 11988811135681198921199041113578 + hellip119888119899119892119904119899 119892119904119894(119909) ∶= 119892(119904119894119909)Indeed there is an open subset 119880 sub 119866 such that inf119880 119892 gt 0 now cover Supp(119891) by finitely many1199041113568119880hellip 119904119899119880

Given 119891 119892 as before define (119891 ∶ 119892) to be the infimum of 1198881113568+hellip+119888119899 among all choices of (1198881113568 hellip 1199041113568 hellip)satisfying the bound above We contend that

(119891119904 ∶ 119892) = (119891 ∶ 119892) 119904 isin 119866 (11)(119905119891 ∶ 119892) = 119905(119891 ∶ 119892) 119905 isin ℝge1113567 (12)

(1198911113568 + 1198911113569 ∶ 119892) le (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) (13)

(119891 ∶ 119892) ge sup 119891sup 119892 (14)

(119891 ∶ ℎ) le (119891 ∶ 119892)(119892 ∶ ℎ) ℎ 119892 isin 119862119888(119866)lowast+ (15)

0 lt 1(1198911113567 ∶ 119891)

le (119891 ∶ 119892)(1198911113567 ∶ 119892)

le (119891 ∶ 1198911113567) 119891 1198911113567 119892 isin 119862119888(119866)lowast+ (16)

and that for all 1198911113568 1198911113569 ℎ isin 119862119888(119866)+ with ℎ|Supp(1198911113578+1198911113579) ge 1 and all 120598 gt 0 there is a compact neighborhood119870 ni 1 such that

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1198911113568 + 1198911113569 ∶ 119892) + 120598(ℎ ∶ 119892) 119892 isin 119862119888(119866 119870)lowast+ (17)

The properties (11) mdash (14) are straightforward For (15) note that

119891 le 1114012119894119888119894119892119904119894 119892 le 1114012

119895119889119895ℎ119905119895 ⟹ 119891 le1114012

119894119895119888119894119889119895ℎ119905119895119904119894

7

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

The group ℝtimesgt1113567 acts on the set of left (resp right) Haar measures by rescaling For commutative

groups we make no distinction of left and right

Example 117 If 119866 is discrete the counting measure Count(119864) = |119864| is a left and right Haar measureWhen 119866 is finite it is customary to take the normalized version 120583 ∶= |119866|minus1113568Count

Definition 118 Let 119866 be a locally compact For 119891 isin 119862119888(119866) we write 119891 ∶ 119909 ↦ 119891(119909minus1113568) 119891 isin 119862119888(119866)For any Radon measure 120583 on 119866 let be the Radon measure with d(119909) = d120583(119909minus1113568) in terms of linearfunctionals 119868(119891) = 119868120583( 119891)

Lemma 119 In the situation above 120583 is a left (resp right) Haar measure if and only if is a right(resp left) Haar measure

Proof An instance of transport of structure because 119909 ↦ 119909minus1113568 is an isomorphism of locally compactgroups 119866 simminusrarr 119866op

Theorem 120 (A Weil) For every locally compact group 119866 there exists a left (resp right) Haar mea-sure on 119866 They are unique up to ℝtimes

gt1113567-action

Proof The following arguments are taken from Bourbaki [1 VII sect12] Upon replacing 119866 by 119866op itsuffices to consider the case of left Haar measures For the existence part we seek a left 119866-invariantpositive linear functional on 119862119888(119866)+ Write

119862119888(119866)lowast+ ∶= 119862119888(119866)+ ∖ 0

For any compact subset 119870 sub 119866 define 119862119888(119866 119870)+ and 119862119888(119866 119870)lowast+ by intersecting 119862119888(119866 119870) with 119862119888(119866)+and 119862119888(119866)lowast+ Observe that for all 119891 isin 119862119888(119866)+ and 119892 isin 119862119888(119866)lowast+ there exist 119899 ge 1 1198881113568 hellip 119888119899 isin ℝge1113567 and1199041113568 hellip 119904119899 isin 119866 such that

119891 le 11988811135681198921199041113578 + hellip119888119899119892119904119899 119892119904119894(119909) ∶= 119892(119904119894119909)Indeed there is an open subset 119880 sub 119866 such that inf119880 119892 gt 0 now cover Supp(119891) by finitely many1199041113568119880hellip 119904119899119880

Given 119891 119892 as before define (119891 ∶ 119892) to be the infimum of 1198881113568+hellip+119888119899 among all choices of (1198881113568 hellip 1199041113568 hellip)satisfying the bound above We contend that

(119891119904 ∶ 119892) = (119891 ∶ 119892) 119904 isin 119866 (11)(119905119891 ∶ 119892) = 119905(119891 ∶ 119892) 119905 isin ℝge1113567 (12)

(1198911113568 + 1198911113569 ∶ 119892) le (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) (13)

(119891 ∶ 119892) ge sup 119891sup 119892 (14)

(119891 ∶ ℎ) le (119891 ∶ 119892)(119892 ∶ ℎ) ℎ 119892 isin 119862119888(119866)lowast+ (15)

0 lt 1(1198911113567 ∶ 119891)

le (119891 ∶ 119892)(1198911113567 ∶ 119892)

le (119891 ∶ 1198911113567) 119891 1198911113567 119892 isin 119862119888(119866)lowast+ (16)

and that for all 1198911113568 1198911113569 ℎ isin 119862119888(119866)+ with ℎ|Supp(1198911113578+1198911113579) ge 1 and all 120598 gt 0 there is a compact neighborhood119870 ni 1 such that

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1198911113568 + 1198911113569 ∶ 119892) + 120598(ℎ ∶ 119892) 119892 isin 119862119888(119866 119870)lowast+ (17)

The properties (11) mdash (14) are straightforward For (15) note that

119891 le 1114012119894119888119894119892119904119894 119892 le 1114012

119895119889119895ℎ119905119895 ⟹ 119891 le1114012

119894119895119888119894119889119895ℎ119905119895119904119894

7

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

hence (119891 ∶ ℎ) le sum119894 119888119894sum119895 119889119895 Apply (15) to both 1198911113567 119891 119892 and 119891 1198911113567 119892 to obtain (16)

To verify (17) let 119865 ∶= 1198911113568 + 1198911113569 +120598ℎ1113569 Use the condition on ℎ to define

120593119894(119909) =⎧⎪⎨⎪⎩119891119894(119892)119865(119909) 119909 isin Supp(1198911113568 + 1198911113569)0 otherwise

isin 119862119888(119866)+ (119894 = 1 2)

Furthermore given 120578 gt 0 we may choose the compact neighborhood 119870 ni 1 such that |120593119894(119909) minus 120593119894(119910)| le 120578whenever 119909minus1113568119910 isin 119870 for 119894 = 1 2 One readily verifies that for all 119892 isin 119862119888(119866 119870)+ and 119894 = 1 2

120593119894119892119904 le (120593119894(119904) + 120578) sdot 119892119904 119904 isin 119866

Suppose that 119865 le 11988811135681198921199041113578 + hellip 119888119899119892119904119899 then 119891119894 = 120593119894119865 le sum119899119895=1113568 119888119895(120593119894(119904119895) + 120578)119892

119904119895 by the previous step As1205931113568 + 1205931113569 le 1 we infer that (1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)sum

119899119895=1113568 119888119895 By (13) and (15)

(1198911113568 ∶ 119892) + (1198911113569 ∶ 119892) le (1 + 2120578)(119865 ∶ 119892) le (1 + 2120578) 1114101(1198911113568 + 1198911113569 ∶ 119892) +1205982(ℎ ∶ 119892)

1114104

le (1198911113568 + 1198911113569 ∶ 119892) + 11141011205982 + 2120578(1198911113568 + 1198911113569 ∶ ℎ) + 120598120578

1114104 (ℎ ∶ 119892)

Choosing 120578 small enough relative to 1198911113568 1198911113569 ℎ 120598 yields (17)Proceed to the construction of Haar measure We fix 1198911113567 isin 119862119888(119866)lowast+ and set

119868119892(119891) ∶=(119891 ∶ 119892)(1198911113567 ∶ 119892)

119891 isin 119862119888(119866)+ 119892 isin 119862119888(119866)lowast+

We want to ldquotake the limitrdquo over 119892 isin 119862119888(119866 119870)lowast+ where 119870 shrinks to 1 and 119891 1198911113567 isin 119862119888(119866)lowast+ are keptfixed By (16) we see 119868119892(119891) isin image ∶= 1114094(1198911113567 ∶ 119891)minus1113568 (119891 ∶ 1198911113567)1114097 For each compact neighborhood 119870 ni 1 in 119866let 119868119870 (119891) ∶= 1114106119868119892(119891) ∶ 119892 isin 119862119888(119866 119870)lowast+1114109 The family of all 119868119870 (119891) form a filter base in the compact space imagehence can be refined into an ultrafilter which has the required limit 119868(119891)

We refer to [2 I sect6 and sect91] for the language of filters and its relation with compactness Alterna-tively one can argue using the MoorendashSmith theory of nets together with Tychonoffrsquos theorem see [6(1525)]

Then 119868 ∶ 119862119888(119866)+ rarr ℝge1113567 satisfies119866-invariance by (11) From (13) we infer 119868(1198911113568+1198911113569) le 119868(1198911113568)+119868(1198911113569)which already holds for all 119868119892 from (17) we infer 119868(1198911113568) + 119868(1198911113569) le 119868(1198911113568 + 1198911113569) + 120598119868(ℎ) whenever ℎ ge 1 onSupp(1198911113568 + 1198911113569) and 120598 gt 0 (true by using 119892 with sufficiently small support) so 119868(1198911113568 + 1198911113569) = 119868(1198911113568) + 119868(1198911113569)follows The behavior under dilation follows from (12) All in all 119868 is the required Haar measure on 119866

Now turn to the uniqueness Let us consider a left (resp right) Haar measure 120583 (resp 120584) on 119866 Itsuffices by Lemma 119 to show that 120583 and are proportional Fix 119891 isin 119862119888(119866) such that int

119866119891 d120583 ne 0 It

is routine to show that the function

119863119891 ∶ 119909⟼ 11141021114009119866119891 d1205831114105

minus1113568

1114009119866119891(119910minus1113568119909) d120584(119910) 119909 isin 119866

is continuous Let 119892 isin 119862119888(119866) Since (119909 119910) ↦ 119891(119909)119892(119910119909) is in 119862119888(119866 times 119866) Fubinirsquos theorem implies that

1114009119866119891 d120583 sdot 1114009

119866119892 d120584 = 1114009

119866119891(119909) 1114102 1114009

119866119892(119910) d120584(119910)1114105 d120583(119909) d120584(119910)=d120584(119910119909)=

11140091198661114009119866119891(119909)119892(119910119909) d120583(119909) d120584(119910) d120583(119909)=d120583(119910119909)= 1114009

11986611141021114009

119866119891(119910minus1113568119909)119892(119909) d120583(119909)1114105 d120584(119910)

= 1114009119866119892(119909) 11141021114009

119866119891(119910minus1113568119909) d120584(119910)1114105 d120583(119909) = 1114009

119866119891 d120583 sdot 1114009

119866119892119863119891 d120583(119909)

8

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Hence int119866119892 d120584 = int

119866119863119891 sdot 119892 d120583 Since 119892 is arbitrary and 119863119891 is continuous 119863119891(119909) is independent of 119891

hereafter written as 119863(119909) Now119863(1)1114009

119866119891 d120583 = 1114009

119866119891 d

for 119891 isin 119862119888(119866) with int119891 d120583 ne 0 As both sides are linear functionals on the whole 119862119888(119866) the equalityextends and yield the asserted proportionality

Proposition 121 Let 120583 be a left (resp right) Haar measure on 119866 Then 119866 is discrete if and only if120583(1) gt 0 and 119866 is compact if and only if 120583(119866) lt +infin

Proof The ldquoonly ifrdquo parts are easy for discrete 119866 we may take 120583 = Count and for compact 119866 weintegrate the constant function 1

For the ldquoifrdquo part first suppose that 120583(1) gt 0 Then every singleton has measure 120583(1) Observethat every compact neighborhood 119870 of 1 must satisfy 120583(119870) gt 0 therefore 119870 is finite As 119866 is Hausdorff1 is thus open so 119866 is discrete

Next suppose 120583(119866) is finite Fix a compact neighborhood 119870 ni 1 so that 120583(119870) gt 0 If 1198921113568 hellip 119892119899 isin 119866are such that 119892119894119870119899119894=1113568 are disjoint then 119899120583(119870) = 120583(⋃119894 119892119894119870) le 120583(119866) and this gives 119899 le 120583(119866)120583(119870)Choose a maximal collection 1198921113568 hellip 119892119899 isin 119866 with the disjointness property above Every 119892 isin 119866 mustlie in 119892119894119870 sdot 119870minus1113568 for some 1 le 119894 le 119899 since maximality implies 119892119870 cap 119892119894119870 ne empty for some 119894 Hence119866 = ⋃119899

119894=1113568 119892119894119870 sdot 119870minus1113568 is compact by 13

15 The modulus characterLet 120579 ∶ 1198661113568 rarr 1198661113569 be an isomorphism of locally compact groups A Radon measure 120583 on 1198661113568 transportsto a Radon measure 120579lowast120583 on 1198661113569 the corresponding positive linear functional is

119868120579lowast120583(119891) = 119868120583(119891120579) 119891120579 ∶= 119891 ∘ 120579 119891 isin 119862119888(1198661113569)

It is justified to express this as d(120579lowast120583)(119909) = d120583(120579minus1113568(119909))Evidently 120579lowast commutes with rescaling by ℝtimes

gt1113567 By transport of structure 120579lowast120583 is a left (resp right)Haar measure if 120583 is For two composable isomorphisms 120579 120590 we have

(120579120590)lowast120583 = 120579lowast(120590lowast120583)

Assume hereafter 1198661113568 = 1198661113569 = 119866 and 120579 isin Aut(119866) We consider 120579minus1113568120583 ∶= (120579minus1113568)lowast120583 where 120583 is a leftHaar measure on 119866 Theorem 120 imlies 120579minus1113568120583 is a positive multiple of 120583 and this ratio does not dependon 120583

Definition 122 For every 120579 isin Aut(119866) define its modulus 120575120579 as the positive number determined by

120579minus1113568120583 = 120575120579120583 equivalently d120583(120579(119909)) = 120575120579 d120583(119909)

where 120583 is any left Haar measure on 119866

Remark 123 One can also use right Haar measures to define the modulus Indeed the right Haarmeasures are of the form form where 120583 is a left Haar measure and

d(120579(119909)) = d120583(120579(119909)minus1113568) = d120583(120579(119909minus1113568)) = 120575120579 sdot d120583(119909minus1113568) = 120575120579 sdot d(119909)

Remark 124 The modulus characters are sometimes defined as Δ119866(119892) = 120575119866(119892)minus1113568 in the literature suchas [1 VII sect13] or [6] This ambiguity is responsible for uncountably many headaches Our conventionfor 120575119866 seems to conform with most papers in representation theory

9

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Proposition 125 The map 120579 ↦ 120575120579 defines a homomorphism Aut(119866) rarr ℝtimesgt1113567

Proof Use the fact that

1114100(120579120590)minus11135681114103lowast120583 = 120590minus1113568lowast (120579minus1113568lowast 120583) = 120575120579 sdot 120590minus1113568lowast 120583 = 120575120579120575120590 sdot 120583

for any two automorphisms 120579 120590 of 119866 and any left Haar measure 120583

Example 126 If 119866 is discrete then 120575120579 = 1 for all 120579 since the counting measure is preserved

Definition 127 For 119892 isin 119866 let Ad(119892) be the automorphism 119909 ↦ 119892119909119892minus1113568 of 119866 Define 120575119866(119892) ∶= 120575Ad(119892) isinℝtimesgt1113567 In other words d120583(119892119909119892minus1113568) = 120575119866(119892) d120583(119909)

The following result characterizes 120575119866

Lemma 128 The map 120575119866 ∶ 119866 rarr ℝtimesgt1113567 is a continuous homomorphism For every left (resp right) Haar

measure 120583ℓ (resp 120583119903) we have

d120583ℓ(119909119892) = 120575119866(119892)minus1113568 d120583ℓ(119909) d120583119903(119892119909) = 120575119866(119892) d120583119903(119909)

Furthermore ℓ = 120575119866 sdot 120583ℓ and 119903 = 120575minus1113568119866 120583119903

Proof Since Ad(sdot) ∶ 119866 rarr Aut(119866) is a homomorphism so is 120575119866 by Proposition 125 As for the continuityof 120575119866 fix a left Haar measure 120583 and 119891 isin 119862119888(119866) with 120583(119891) ∶= int

119866119891 d120583 ne 0 Then

120575119866(119892)minus1113568 = 120583(119891)minus11135681114009119866119891 ∘ Ad(119892) d120583

and it is routine to check that int119866119891 ∘ Ad(119892) d120583 is continuous in 119892 The second assertion results from

applying d120583ℓ(sdot) and d120583119903(sdot) to

119909119892 = 119892(Ad(119892minus1113568)119909) 119892119909 = (Ad(119892)119909)119892

respectivelyTo deduce the last assertion note that 120575119866120583ℓ is a right Haar measure by the previous step and Remark

115 thus Theorem 120 entailsexist119905 isin ℝtimes

gt1113567 120583ℓ = 119905120575119866120583ℓ

Hence 120583ororℓ = 119905120575minus1113568119866 120583ℓ = 1199051113569120583ℓ and we obtain 119905 = 1 A similar reasoning for 120575minus1113568119866 120583119903 applies

Corollary 129 We have 120575119866 = 1 if and only if every left Haar measure on119866 is also a right Haar measureand vice versa

Proposition 130 For every 119892 isin 119866 we have 120575119866(119892)minus1113568 = 120575119866op(119892)

Proof Fix a left Haar measure 120583 for 119866 and denote the multiplication in 119866op by ⋆ Then d120583(119892119909119892minus1113568) =120575119866(119892) d120583(119909) is equivalent to d120583(119892minus1113568 ⋆ 119909 ⋆ 119892) = 120575119866(119892) d120583(119909) for 119866op Since 120583 is a right Haar measure for119866op we infer from Remark 123 that 120575119866op(119892minus1113568) = 120575119866(119892)

Definition 131 A locally compact group 119866 with 120575119866 = 1 is called unimodular

Example 132 The following groups are unimodular

bull Commutative groups

bull Discrete groups use Example 126

10

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

bull Compact groups indeed the compact subgroup 120575119866(119866) of ℝtimesgt1113567 ≃ ℝ must be trivial

The normalized absolute value on a local field 119865 is actually the modulus character for the additivegroup of 119865 as shown below

Theorem 133 For every local field 119865 define sdot 119865 to be the normalized absolute value | sdot |119865 for non-archimedean 119865 the usual absolute value for 119865 = ℝ and 119909 ↦ |119909|ℝ for 119865 = ℂ For 119905 isin 119865times let 119898119905 be theautomorphism 119909 ↦ 119905119909 for (119865 +) Then

120575119898119905 = 119905119865

Proof Since 119898119905119905prime = 119898119905prime ∘ 119898119905 by Proposition 125 we have 120575119898119905119905prime = 120575119898119905prime120575119898119905 Fix any Haar measure 120583 on119865 By definition

120575119898119905 =120583(119905119870)120583(119870)

for every compact subset 119870 sub 119865 The assertion 120575119898119905 = 119905119865 is then evident for 119865 = ℝ or ℂ by taking 119870to be the unit ball and work with the Lebesgue measure

Now suppose 119865 is non-archimedean and take 119870 = 120108119865 For 119905 isin 120108times119865 we have 119905119870 = 119870 thus 120575119898119905 = 1 Thesame holds for | sdot |119865 and both are multiplicative It remains to show 120575119898120603 = |120603|119865 = 119902minus1113568 for any uniformizer120603 of 119865 where 119902 ∶= |120108119865120109119865 | But then 120583(120603120108119865)120583(120108119865) = (120108119865 ∶ 120603120108119865)minus1113568 = 119902minus1113568

16 Interlude on analytic manifoldsIt is well known that every real Lie group admits an analytic structure The theory of analytic manifoldsgeneralizes to any local field 119865 For a detailed exposition we refer to [9 Part II]

In what follows it suffices to assume that 119865 is a complete topological field with respect to someabsolute value |sdot| Let119880 sub 119865119898 be some open ball The definition of 119865-analytic functions 119891 = (1198911113568 hellip 119891119899) ∶119880 rarr 119865119899 is standard we require that each 119891119894 can be expressed as convergent power series on 119880 For anyopen subset 119880 sub 119865119898 we say 119891 ∶ 119880 rarr 119865119899 is 119865-analytic if 119880 can be covered by open balls over which 119891are 119865-analytic

Definition 134 An 119865-analytic manifold of dimension 119899 is a second countable Hausdorff space 119883 en-dowed with an atlas 1114106(119880 120601119880 ) ∶ 119880 isin 1199841114109 where

bull 119984 is an open covering of 119883

bull for each 119880 isin 119984 120601119880 is a homeomorphism from 119880 to an open subset 120601119880 (119880) of 119865119899

The condition is that for all 1198801113568 1198801113569 isin 119984 with 1198801113568 cap 1198801113569 ne empty the transition map

1206011198801113579120601minus11135681198801113578 ∶ 1206011198801113578(1198801113568 cap 1198801113569) rarr 1206011198801113579(1198801113568 cap 1198801113569)

is 119865-analytic

As in the standard theory of manifolds the 119865-analytic functions on 119883 are defined in terms of chartsand the same applies to morphisms between 119865-analytic manifolds the notion of closed submanifolds isdefined in the usual manner We can also pass to maximal atlas in the definition The 119865-analytic manifoldsthen form a category

Open subsets of 119865-analytic manifolds inherit analytic structures It is also possible to define 119865-analyticmanifolds intrinsically by sheaves

More importantly the notion of tangent bundles cotangent bundles differential forms and their ex-terior powers still make sense on an 119865-analytic manifold 119883 Let 119899 = dim119883 As in the real case locallya differential form on 119883 can be expressed as 119888(1199091113568 hellip 119909119899) d1199091113568 and⋯ and d119909119899 where 1114100119880 120601119880 = (1199091113568 hellip 119909119899)1114103is any chart for 119883 and 119888 is an analytic function on 120601119880 (119880) sub 119865119899

11

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Remark 135 When (119865 | sdot |) is ultrametric the theory of 119865-analytic manifolds is not quite interestingsince they turn out to be totally disconnected Nevertheless it is a convenient vehicle for talking aboutmeasures exponential maps and so forthExample 136 Let X be an 119865-variety The set 119883 of its 119865-points acquires a natural Hausdorff topology(say by reducing to the case to affine varieties then to affine sets) If X is smooth then 119883 becomes ananalytic 119865-manifold The assignment X ↦ 119883 is functorial The algebraically defined regular functionsdifferentials etc on X give rise to their 119865-analytic avatars on 119883 By convention objects on X will oftencarry the adjective algebraic

Now assume that 119865 is local and fix a Haar measure on 119865 denoted abusively by d119909 in conformity withthe usual practice This equips 119865119899 with a Radon measure for each 119899 ge 0 and every open subset 119880 sub 119865119899carries the induced measure

Let 119883 be an 119865-analytic manifold of dimension 119899 The collection of volume forms will be an ℝtimesgt1113567-

torsor over 119883 Specifically a volume form on 119883 is expressed in every chart (119880 120601119880 ) as

|120596| = 119891(1199091113568 hellip 119909119899)| d1199091113568|⋯ | d119909119899|where 119891 ∶ 119881 ∶= 120601119880 (119880) rarr ℝge1113567 is continuous Define sdot 119865 ∶ 119865 rarr ℝge1113567 as in Theorem 133 A change oflocal coordinates 119910119894 = 119910119894(1199091113568 hellip 119909119899) (119894 = 1hellip 119899) transports 119891 and has the effect

| d1199101113568|⋯ | d119910119899| =11140651114065det 1114102

120597119910119894120597119909119895

1114105119894119895

11140651114065119865

sdot | d1199091113568|⋯ | d119909119899| (18)

In other words |120596| is the ldquoabsolute valuerdquo of a differential form of top degree on119883 squared when 119865 = ℂSome geometers name these objects as densities

Now comes integration The idea is akin to the case of 119862infin-manifolds and we refer to [7 Chapter16] for a complete exposition for the latter Let 120593 isin 119862119888(119883) and choose an atlas (119880 120601119880 ) ∶ 119880 isin 119984 for119883 In fact it suffices to take a finite subset of 119984 covering Supp(120593)

1 By taking a partitions of unity 1114100120595119880 ∶ 119883 rarr [0 1]1114103119880isin119984

(with sum119880 120595119880 = 1 as usual) and replacing120601 by 120595119880120601 for each 119880 isin 119984 the definition of int

119883120593|120596| reduces to the case that Supp(120593) sub 119880 for

some 119880 isin 119984

2 In turn we are reduced to the integration int119881sub119865119899

120593119891 d1199091113568⋯ d119909119899 on some open subset 119881 sub 119865119899 thiswe can do by the choice of Haar measure on 119865 Furthermore this integral is invariant under 119865-analytic change of coordinates Indeed this is a consequence of (18) plus the interpretation of sdot 119865 in Theorem 133

3 As in the case overℝ one verifies thatint120593120596 does not depend on the choice of charts and partitionof unity

4 It is routine to show that 119868 ∶ 120593 ↦ int119883120593|120596| is a positive linear functional giving rise to a Radon

measure on 119883 The operations of addition rescaling and group actions on volume forms mirrorthose on Radon measures

Remark 137 In many applications 119883 will come from a smooth 119865-variety X or its open subsets and|120596| will come from a nowhere-vanishing algebraic differential form 120596 thereon say by ldquotaking absolutevalues sdot 119865rdquo we call such an 120596 an algebraic volume form on XDefinition 138 Let 119883 be an 119865-analytic manifolds where 119865 is a local field Set

119862infin119888 (119883) ∶=⎧⎪⎨⎪⎩1114106119891 isin 119862119888(119883) ∶ infinitely differentiable1114109 archimedean 1198651114106119891 isin 119862119888(119883) ∶ locally constant1114109 non-archimedean 119865

where locally constant means that every 119909 isin 119883 admits an open neighborhood119880 such that 119891|119880 is constant

12

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

17 More examplesWe begin by pinning down the Haar measures on a local field 119865 viewed as additive groups

Archimedean case 119865 = ℝ Use the Lebesgue measure

Archimedean case 119865 = ℂ Use twice of the Lebesgue measure

Non-archimedean case To prescribe a Haar measure on 119865 by Theorem 120 it suffices to assign a vol-ume to the compact subring 120108119865 of integers A canonical choice is to stipulate that 120108119865 has volume1 but this is not always the best recipe

Example 139 If 119866 is a Lie group over any local field 119865 ie a group object in the category of 119865-analyticmanifolds one can construct left (resp right) Haar measures as follows Choose a basis 1205961113568 hellip 120596119899 ofthe cotangent space of 119866 at 1 The cotangent bundle of 119866 is trivializable by left (resp right) translationtherefore 1205961113568 hellip 120596119899 can be identified as globally defined left (resp right) invariant differentials Thenowhere-vanishing top form 120596 ∶= 1205961113568 and⋯ and 120596119899 or more precisely |120596| gives the required measure on119866 The Haar measures constructed in this way are unique up to scaling as the form 120596 (thus |120596|) is

We shall denote the tangent space at 1 isin 119866 as 120100 we have dim119865 120100 = 119899 The automorphism Ad(119892) ∶119909 ↦ 119892119909119892minus1113568 induces a linear map on 120100 at 1 denoted again by Ad(119892)

Proposition 140 In the circumstance of Example 139 one has

120575119866(119892) = 1114062 det(Ad(119892) ∶ 120100 rarr 120100) 1114062119865 119892 isin 119866

Proof This follows from Theorem 133

Example 141 Let 119865 be a local field For the multiplicative group 119865times the algebraic volume form

119909minus1113568 d119909

defines a Haar measure it is evidently invariant under multiplications More generally let119866 = GL(119899 119865)We take 120578 to be a translation-invariant nonvanishing algebraic volume form on 119872119899(119865) ≃ 1198651198991113579 The 119866-invariant algebraic volume form

120596(119892) ∶= (det 119892)minus119899120578(119892) 119892 isin 119866

induces a left and right Haar measure on 119866

Example 142 Here is a non-unimodular group Let 119865 be a local field and 119899 = 1198991113568 + ⋯ + 1198991113569 where1198991113568 1198991113569 isin ℤge1113568 Consider the group of block upper-triangular matrices

119875 ∶= 1114108119892 = 11141021198921113568 119909

11989211135691114105isin GL(119899 119865) ∶ 119892119894 isin GL(119899119894 119865) 119909 isin 1198721198991113578times1198991113579(119865)1114111

the unspecified matrix entries being zero This is clearly the space of 119865-points of an algebraic group PThe Lie algebra of P is simply ( lowast lowastlowast ) A routine calculation of block matrices gives

11141021198921113568 119909

11989211135691114105 1114102119886 119887

1198891114105 11141021198921113568 119909

11989211135691114105minus1113568

= 1114102119886 1198921113568119887119892minus11135681113569

119889 1114105

where 119886 isin 1198721198991113578(119865) 119887 isin 1198721198991113579(119865) and 119887 isin 1198721198991113578times1198991113579(119865) Thus Proposition 140 and Theorem 133 imply

120575119875(119892) = det 11989211135681198991113579119865 sdot det 1198921113569

minus1198991113578119865

13

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

18 Homogeneous spacesLet 119866 be a locally compact group In what follows we consider spaces with right 119866-actions One canswitch to left 119866-actions by passing to 119866op

Definition 143 A 119866-space in the category of locally compact spaces is a locally compact Hausdorffspace 119883 equipped with a continuous right 119866-action 119883 times119866 rarr 119883 If the 119866-action is transitive 119883 is calleda homogeneous 119866-space

The 119866-spaces form a category the morphisms are continuous 119866-equivariant maps

Example 144 Let119867 be a closed subgroup of119866 The coset space119867119866 is locally compact and Hausdorffby Proposition 14 and 119866 acts continuously on the right by (119867119909 119892) ↦ 119867119909119892 This is a homogeneous 119866-space

For a general homogeneous 119866-space 119883 and 119909 isin 119883 let 119866119909 ∶= Stab119866(119909) it is a closed subgroup andwe have 119866119909119892 = 119892minus1113568119866119909119892 for 119892 isin 119866 The orbit map orb119909 ∶ 119892 ↦ 119909119892 factors through

orb119909 ∶ 119866119909119866 rarr 119883119866119909119892⟼ 119909119892

which is a continuous 119866-equivariant bijection If orb119909 is actually an isomorphism then 119883 ≃ 119866119909119866 inthe category of 119866-spaces This indeed holds under mild conditions on 119866 thereby showing that Example144 is the typical sort of homogeneous spaces

Proposition 145 Suppose that119866 is second countable as a topological space Then for any homogeneous119866-space 119883 and any 119909 isin 119883 the map orb119909 ∶ 119866119909119866 rarr 119883 is an isomorphism of 119866-spaces

Proof It suffices to show that orb119909 is an open map Let119880 ni 1 be any compact neighborhood in119866 Thereis a sequence 1198921113568 1198921113569 hellip isin 119866 such that 119866 = ⋃119894ge1113568119880119892119894 therefore 119883 = ⋃119894ge1113568 119909119880119892119894 Each 119909119880119892119894 is compacthence closed in 119883

Bairersquos theorem implies that some 119909119880119892119894 has nonempty interior Since 119909119880119892119894 ≃ 119909119880 by right translationwe obtain an interior point 119909ℎ of the subset 119909119880 of 119883 with ℎ isin 119880 Then 119909 is an interior point of 119909119880ℎminus1113568hence of the bigger subset 119909119880119880minus1113568

Now consider any open subset 119881 sub 119866 and 119892 isin 119881 Take a compact neighborhood 119880 ni 1 such that119880119880minus1113568119892 sub 119881 Then

119909119892 isin 119909119880119880minus1113568119892 sub 119909119881As 119909 has been shown to an interior point of 119909119880119880minus1113568 so is 119909119892 in 119909119880119880minus1113568119892 sub 119909119881 Since 119892 119881 are arbitrarywe conclude that 119892 ↦ 119909119892 is an open map The same holds for orb119909 by the definition of quotient topologies

Corollary 146 If119866 is an 119865-analytic Lie group where 119865 is a local field then the condition in Proposition145 holds thus every homogeneous 119866-space takes the form 119867119866 for some 119867

Proof The second countability is built into the definitions

Example 147 Take 119883 to be the (119899 minus 1)-sphere 120138119899minus1113568 on which the orthogonal group O(119899ℝ) acts con-tinuously (in fact analytically or even algebraically) and transitively The action is given by (119907 119892) ↦ 119907119892where we regard 119907 as a row vector The subgroup SO(119899ℝ) also acts transitively on 120138119899minus1113568 since everyvector 119907 isin 120138119899minus1113568 is fixed by reflections relative to any hyperplane containing 119907 which have determinantminus1 This gives

120138119899minus1113568 ≃ O(119899 minus 1ℝ)O(119899ℝ)Indeed the stabilizer group of 119907 = (1 0 hellip 0) can be identified with O(119899 minus 1ℝ)

14

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Example 148 Another important example of homogeneous space is given by the Siegel upper halfplane Here it is customary to work with left actions by setting

Sp(2119899ℝ) ∶= 1114106119892 isin GL(2119899ℝ) ∶ 119892 sdot 119869 sdot 119905119892 = 1198691114109 119869 ∶= 1114102minus1119899times119899

1119899times119899 1114105

In a coordinate-free language Sp(2119899ℝ) is the isometry group of the symplectic form prescribed by 119869 Take 119883 to be

ℋ119899 ∶= 119885 = 119877 + 119894119878 isin 119872119899times119899(ℂ) ∶ 119877 119878 isin 119872119899times119899(ℝ) symmetric 119878 gt 0

Let Sp(2119899ℝ) act on the left of ℋ119899 by

119892119885 ∶= (119860119885 + 119861)(119862119885 + 119863)minus1113568 119885 isin ℋ119899

119892 = 1114102119860 119861119862 1198631114105 isin Sp(2119899ℝ) 119860 119861 119862119863 isin 119872119899times119899(ℝ)

This is manifestly an analytic map in (119892 119885) One can show that

bull this is indeed a transitive group action and

bull the stabilizer group of 119885 = 119894 sdot 1119899times119899 is the unitary group 119880(119899)

Therefore ℋ119899 ≃ Sp(2119899ℝ)119880(119899) When 119899 = 1 it reduces to the usual ℋ1113568 ∶= 120591 isin ℂ ∶ image(120591) gt 0 withSL(2ℝ) acting by linear fractional transformations

Example 149 Let 119865 be a non-archimedean local field with ring of integers 120108119865 By convention weidentify 119865119899 with the space of row vectors and let GL(119899 119865) act from the right

A lattice in an 119899-dimensional 119865-vector space 119881 is a free 120108119865-submodule 119871 sub 119881 of rank 119899 such that119871 sdot 119865 = 119881 the standard example is 120108119899119865 inside 119865119899 Define the discrete space

119883 ∶= lattices 119871 sub 119881

with right GL(119881)-action Exercise check that the GL(119881)-action is continuous and transitive in factStabGL(119881)(119871) is closed and open in GL(119881)

Now identify 119881 with 119865119899 by choosing a basis The stabilizer of 119871 = 120108119899119865 is GL(119899 120108119865) hence

119883 ≃ GL(119899 120108119865)GL(119899 119865)

Examples 147mdash148 are well-known instances of Riemannian symmetric spaces Loosely speakingthe space119883 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the sametype as Example 148 it consists of the vertices of the enlarged BruhatndashTits building of GL(119881)

Next we turn to the measures on the 119866-spaces 119867119866 where 119867 is a closed subgroup of 119866

Lemma 150 Fix a right Haar measure 120583119867 on 119867 The map

119862119888(119866)⟶ 119862119888(119867119866)

119891⟼ 119891 ∶= 1114096119867119892 ↦ 1114009119867119891(ℎ119892) d120583119867 (ℎ)1114099

is well-defined surjective and 119866-equivariant with respect to the left 119866-actions on 119862119888(119866) and 119862119888(119867119866)given by 119891 ↦ 1114094119892119891 ∶ 119909 ↦ 119891(119909119892)1114097 It maps 119862119888(119866)+ onto 119862119888(119867119866)+

15

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Proof The only nontrivial part is the surjectivity Given 119891 isin 119862119888(119867119866) by Lemma 15 there exists acompact 119870 sub 119866 such that 119892 ↦ 119891(119867119892) is supported on 119867119870 We want to define

119891(119892) ∶=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

119891(119867119892) sdot 120601(119892)120601(119867119892) 119892 isin 119867119870

0 119892 notin 119867119870

for some 120601 isin 119862119888(119866)+ depending solely on 119870 Specifically take any 120601 isin 119862119888(119866)+ such that 120601|119870 gt 0 Forany 119892 isin 119867119870 we have

120601(119892) ge inf119909isin119870

120601(119867119909) gt 0

by assumption hence 119891 is a well-defined element in 119862119888(119866) If 119891 isin 119862119888(119867119866)+ then 119891 isin 119862119888(119866)+It is also readily verified that int

119867119891(ℎ119892) d120583119867 (ℎ) = 119891(119867119892) for 119892 isin 119867119870 and zero for 119892 notin 119867119870 this

affords the required preimage of 119891

Fix a right Haar measure 120583119867 on 119867 to define 119891 ↦ 119891 as in Lemma 150 Every positive Radonmeasure 120583119867119866 on 119883 induces 120583119867119866 on 119866 characterized by int

119866119891 d120583119867119866 = int119867119866 119891

d120583119867119866Also recall the notion of quasi-invariance from Definition 114

Lemma 151 Fix a right Haar measure 120583119867 on 119867 and let 120583119867119866 be a quasi-invariant positive measureon 119883 with eigencharacter 120594 ∶ 119866 rarr ℝtimes

gt1113567 Then 120583119867119866 is of eigencharacter 120594 (resp 120575119867 (ℎ)) under right119866-action (resp left 119867-action)

Proof Since 119891 ↦ 119891 respects right 119866-translation the eigencharacter of 120583119867119866 is the same 120594 Now let119891 isin 119862119888(119866) recall that 119891ℎ(119892) = 119891(ℎ119892) By Lemma 128 (119891ℎ)(119867119892) equals

1114009119867119891ℎ(119896119892) d120583119867 (119896) = 1114009

119867119891(ℎ119896119892) d120583119867 (119896) = 120575119867 (ℎ)minus11135681114009

119867119891(ℎprime119892) d120583119867 (ℎprime)

which is 120575119867 (ℎ)minus1113568119891(119867119892) This implies that int119866119891ℎ d120583119867119866 = 120575119867 (ℎ)minus1113568int119866 119891 d120583119867119866 as asserted

Theorem 152 Let 120594 ∶ 119866 rarr ℝtimesgt1113567 be a continuous homomorphism Then there exists a quasi-invariant

positive measure on 119867119866 of eigencharacter 120594 if and only if

120594|119867 = 120575119867 (120575119866|119867 )minus1113568

Moreover such a quasi-invariant measure is unique up toℝtimesgt1113567 when 120594|119867 = 120575119867120575119866|119867 and every choice of

right Haar measures 120583119866 (resp 120583119867 ) on 119866 (resp 119867) gives rise to such a measure 120583119867119866 characterized by

1114009119866119891(119892)120594(119892) d120583119866(119892) = 1114009

11986711986611141021114009

119867119891(ℎ119892) d120583119867 (ℎ)1114105 d120583119867119866(119867119892) (19)

where 119891 isin 119862119888(119866)Consequently 119866-invariant positive measures exists on 119867119866 if and only if 120575119867 = 120575119866|119867 in which case

it is unique up to rescaling

Proof First suppose 120583119867119866 with eigencharacter 120594 is given on 119867119866 Let us verify 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 and

(19) Fix a right Haar measure 120583119867 on 119867 to apply Lemma 151 to obtain 120583119867119866 Lemma 151 implies120583119867119866 = 120594 sdot 120583119866 for a right Haar measure 120583119866 on 119866 Both sides are quasi-invariant under left 119867-actionthe eigencharacter of 120583119867119866 is 120575119867 whilst that of 120594 sdot 120583119866 is (120594120575119866)|119867 by Remark 115 and Lemma 128 Theformula (19) is built into our construction

16

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Conversely given 120594 with 120594|119867 = 120575119867 (120575119866|119867 )minus1113568 fix 120583119866 and 120583119867 We claim that

119868 ∶ 119862119888(119867119866)⟶ ℂ

119891 ⟼1114009119866119891120594 d120583119866 119891 ↦ 119891

is well-defined Granting this one readily sees that 119868 defines a Radon measure 120583119867119866 by Lemma 150Furthermore since 119891 ↦ 119891 respects right 119866-action 120583119867119866 will have the same eigencharacter as 120594120583119866namely 120594 What remains to show is that 119891 = 0 implies int

119866119891120594 d120583119866 = 0 For any 120593 isin 119862119888(119866) Fubinirsquos

theorem entails

0 = 1114009119866120593(119892)119891(119867119892)120594(119892) d120583119866(119892) = 1114009

1198661114009119867120593(119892)119891(ℎ119892)120594(119892) d120583119867 (ℎ) d120583119866(119892)

= 11140091198671114009119866120594(119892prime)120593(ℎminus1113568119892prime)119891(119892prime)120594(ℎ)minus1113568120575119866(ℎ)minus1113568 d120583119866(119892prime) d120583119867 (ℎ)

by first swapping the integrals and then substituting 119892prime = ℎ119892 using d120583119866(119892) = 120575119866(ℎ)minus1113568 d120583119866(119892prime) fromLemma 128 Next substitute ℎprime = ℎminus1113568 into the last integral and use d120583119867 (ℎ) = 120575119867 (ℎprime)minus1113568 d120583119867 (ℎprime) fromLemma 128 to arrive at

0 = 11140091198671114009119866120594(119892prime)120593(ℎprime119892prime)119891(119892prime) 120594(ℎprime)120575119866(ℎprime)120575119867 (ℎprime)minus1113568

=1113568

d120583119867 (ℎprime) d120583119866(119892prime)

= 1114009119866120593(119867119892prime)119891(119892prime)120594(119892prime) d120583119866(119892prime)

Select 120593 so that 120593 = 1 on the image of Supp(119891) in 119867119866 to deduce int119866119891120594 d120583119866 = 0

When 119866 is an 119865-analytic Lie group and 119867 is a Lie subgroup the results can also be deduced byconsiderations of volume forms We leave this to the curious reader

Definition 153 Denote by 120583119866120583119867 the quasi-invariant measure on 119867119866 of eigencharacter 120594 determinedby (19) To see the benefits of this notation note that (119904120583119866)(119905120583119867 ) = (119904119905) sdot 120583119866120583119867 for all 119904 119905 isin ℝtimes

gt1113567

Remark 154 For homogeneous spaces under left 119866-actions one uses left Haar measures on119867 to define119891 ↦ 119891 and the condition in Theorem 152 becomes

120594|119867 = (120575119867 )minus1113568 sdot 120575119866|119867

Indeed one switches to right 119866op-actions and applies the previous result the eigencharacter 120594 is unal-tered whereas the modulus characters are replaced by inverses by Proposition 130

2 ConvolutionLet 119866 be a locally compact group

21 Convolution of functionsFix a right Haar measure 120583 on 119866 and define 119871119901(119866) ∶= 119871119901(119866 120583) for all 1 le 119901 le infin Write 119891(119909) ∶= 119891(119909minus1113568)(sometimes as 119891or(119909)) for all functions 119891 on 119866

Given two functions 1198911113568 1198911113569 on 119866 their convolution is defined as the function

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910)

= 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910) 119909 isin 119866

(21)

17

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

whenever it converges We will also consider (21) as an element in various 119871119901-spaces on119866 by estimating1198911113568 ⋆ 1198911113569119871119901 in terms of the norms of 1198911113568 and 1198911113569 (see below)Remark 21 If we use left Haar measures instead or equivalently work in 119866op then (21) becomesint1198661198911113568(119909119910)1198911113569(119910minus1113568) d120583(119910) = int119866 1198911113568(119910)1198911113569(119910

minus1113568119909) d120583(119910)

DefinitionndashProposition 22 Let 1198911113568 1198911113569 isin 1198711113568(119866) Then 1198911113568 ⋆ 1198911113569 is a well-defined element in 1198711113568(119866)satisfying

1198911113568 ⋆ 11989111135691198711113578 le 11989111135681198711113578 sdot 11989111135691198711113578

The convolution product makes 1198711113568(119866) into a Banach algebra which is in general non-commutative non-unital

Proof To show that 1198911113568⋆1198911113569 isin 1198711113568(119866) is well-defined evaluate∬119866times119866

|1198911113568(119909119910minus1113568)| sdot |1198911113569(119910)| d120583(119910) d120583(119909) usingFubinirsquos theorem to get 11989111135681198711113579 sdot 11989111135691198711113578 It is straightforward to verify that

1198911113568 ⋆ (1198911113569 ⋆ 1198911113570) = (1198911113568 ⋆ 1198911113569) ⋆ 11989111135701198911113568 ⋆ (1198911113569 + 1198911113570) = 1198911113568 ⋆ 1198911113569 + 1198911113568 ⋆ 1198911113570(1198911113568 + 1198911113569) ⋆ 1198911113570 = 1198911113568 ⋆ 1198911113570 + 1198911113569 ⋆ 1198911113570

Hence (1198711113568(119866) ⋆) is a Banach algebra

It is important to extend the convolution to more general functions and deduce the correspondingestimates Following analystsrsquo convention define a bijection 119901 ↦ 119901prime from [1 +infin] to itself by requiring

1119901 +

1119901prime = 1

Theorem 23 (Minkowskirsquos inequality) Suppose that 1 le 119901 le infin Then for 1198911113568 isin 119871119901(119866) and 1198911113569 isin 1198711113568(119866)(21) is well-defined in 119871119901(119866) and we have

11140621198911113568 ⋆ 11989111135691114062119871119901 le 1198911113568119871119901 sdot 11989111135691198711113578

Proof The case 119901 = 1 has just been addressed and the case 119901 = infin is easy Assume 1 lt 119901 lt infin Toease notations we may and do assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality to the measure 1198911113569(119910) d120583(119910)to obtain

11140091198661198911113568(119909119910minus1113568)1198911113569(119910) d120583(119910) le 11141021114009

1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910)1114105

1113568119901

111410211140091198661198911113569(119910) d120583(119910)1114105

1113568119901prime

Hence

1198911113568 ⋆ 1198911113569119871119901 le 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909119910minus1113568)1199011198911113569(119910) d120583(119910) d120583(119909)1114105

1113568119901

= 11141021198911113569119901minus11135681198711113578 sdot1113962

119866times1198661198911113568(119909)1199011198911113569(119910) d120583(119909) d120583(119910)1114105

1113568119901

= 11141001198911113569119901minus11135681198711113578 sdot 1198911113568

119901119871119901 sdot 119891111356911987111135781114103

1113568119901= 1198911113568119871119901 sdot 11989111135691198711113578

where the right invariance of 120583 is used

18

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Theorem 24 (Youngrsquos inequality) Suppose that 1 le 119901 119902 119903 le infin satisfy

1119902 + 1 =

1119901 +

1119903

Then for all 1198911113568 1198911113569 such that 1198911113568 isin 119871119901(119866) and 1198911113569 isin 119871119903(119866) the convolution (21) is well-defined in 119871119902(119866)and satisfies

1198911113568 ⋆ 1198911113569119871119902 le 1198911113568119871119901 sdot 1198911113569119871119903

Proof The premises imply

1119903prime +

1119902 +

1119901prime = 1

119901119902 +

119901119903prime = 1

119903119902 +

119903119901prime = 1

First assume 1198911113568 1198911113569 ge 0 Apply Houmllderrsquos inequality with three exponents (119903prime 119902 119901prime) and use the rightinvariance of 120583 to deduce that

(1198911113568 ⋆ 1198911113569)(119909) = 11140091198661198911113568(119910minus1113568)1198911113569(119910119909) d120583(119910)

= 11140091198661198911113568(119910minus1113568)

119901119903prime 11141021198911113568(119910minus1113568)

119901119902 1198911113569(119910119909)

119903119902 1114105 1198911113569(119910119909)

119903119901prime d120583(119910)

le 11141021114009119866( 1198911113568)119901 d1205831114105

1113568119903prime

111410211140091198661198911113568(119910minus1113568)119901 sdot 1198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

111410211140091198661198911113569(119910119909)119903 d120583(119910)1114105

1113568119901prime

= 1114062 11989111135681114062119901119903prime

119871119901 111410211140091198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119910)1114105

1113568119902

1198911113569119903119901prime119871119903

It follows from Fubinirsquos theorem that

1198911113568 ⋆ 1198911113569119871119902 le 1114062 11989111135681114062119901119903prime

119871119901 1198911113569119903119901prime119871119903 11141021113962

119866times1198661198911113568(119910minus1113568)1199011198911113569(119910119909)119903 d120583(119909) d120583(119910)1114105

1113568119902

= 1114062 11989111135681114062119901119903prime

119871119901 sdot 1198911113569119903119901prime119871119903 sdot 1198911113568

119901119902119871119901 sdot 1198911113569

119903119902119871119903 = 1198911113568119871119901 sdot 1198911113569119871119903

In general we can work with |1198911113568| |1198911113569| to reduce to the previous setting

Proposition 25 If 120593 isin 119862119888(119866) and 119891 isin 119871119903(119866) (resp 119891 isin 119871119903(119866)) with 1 le 119903 le infin then 120593⋆119891 (resp 119891⋆120593)is defined by a convergent integral and is a continuous function on 119866

Proof Recall 120593119892(119909) ∶= 120593(119892119909) and 119892120593(119909) = 120593(119909119892) for 119892 119909 isin 119866 Apply Theorem 24 with 119902 = infin and1113568119901 +

1113568119903 = 1 to see

|(120593 ⋆ 119891)(119892119909) minus (120593 ⋆ 119891)(119909)| = |((120593 minus 120593119892) ⋆ 119891)(119909)| le 1114062120593or minus (120593119892)or1114062119871119901 11140621198911114062119871119903

We conclude by noting that

1114062120593or minus (120593119892)or1114062119871119901 = 1114063120593or minus 119892minus1113578(120593or)1114063

119871119901119892rarr1113568minusminusminusminusrarr 0

The case with 119891 isin 119871119903(119866) is completely analogous

22 Convolution of measures[ UNDER CONSTRUCTION ]

19

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

3 Continuous representations31 General representationsWe will work exclusively with topological vector spaces over ℂ Some words on Hausdorff propertya topological vector space 119881 is also a topological group under + therefore 119881 being Hausdorff is thesame as 0 = 0 As easily observed 0 sub 119881 is a closed vector subspace thus 1198810 is a Hausdorfftopological vector space The assignment 119881 ↦ 1198810 is functorial and is the left adjoint of the inclusionfunctor

Hausdorff ones topological vector spacesThe main topological vector spaces under consideration are Hausdorff and locally convex the latter

means that there is a basis of neighborhoods consisting of convex subsets Alternatively locally convexHausdorff spaces are exactly the topological vector spaces whose topology is defined by a family ofseparating semi-norms

Definition 31 Henceforth we denote by TopVect the category of locally convex Hausdorff topologicalvector spaces over ℂ In contrast Vect will stand for the category of ℂ-vector spaces

Remark 32 The property of being locally convex and Hausdorff is preserved under passing to subspacesand to quotients by a closed subspace The direct sums of locally convex Hausdorff spaces also carry alocally convex Hausdorff topology characterized in categorical terms

More generally one can form the inductive limits limminusminusrarr inside the category of locally convex spacesthese limits are not necessarily Hausdorff but one can take quotient by 0 to land in TopVect

Example 33 The Hilbert Banach Freacutechet and LF-spaces are all in TopVect Every finite-dimensionalvector space 119881 admits a unique structure of Hausdorff topological vector space namely the usual onecoming from 119881 ≃ ℂdim119881 which is also locally convex

Denote by AutTopVect(119881) (resp AutVect(119881)) the group of automorphisms of 119881 taken in TopVect(resp in the category of ℂ-vector spaces) AutVect(119881) is much larger than AutTopVect(119881) for infinite-dimensional 119881

We will consider certain linear left actions of a locally compact group 119866 on a topological vectorspace 119881 This is the same as prescribing a homomorphism 120587 ∶ 119866 rarr AutVect(119881) of groups it can alsobe recorded by the corresponding ldquoaction maprdquo 119886 ∶ 119866 times 119881 rarr 119881 with 119886(119892 119907) = 119892119907 = 120587(119892)119907 The issue ofcontinuity is more delicate

Definition 34 Let 119866 be a locally compact group and let 119881 in TopVect A continuous representationof 119866 on the left of 119881 is a homomorphism 120587 ∶ 119866 rarr AutVect(119881) such that the corresponding action map

119886 ∶ 119866 times 119881 ⟶ 119881(119892 119907)⟼ 119892119907 ∶= 120587(119892)119907

is continuous The case of right actions is defined analogously with the action written as (119907 119892) ↦ 119907119892with the property 119907(119892119892prime) = (119907119892)119892prime etc

We denote a continuous representation of 119866 as (120587 119881) 119881 or 120587 interchangeably depending on thecontext call 119881 = 119881120587 the underlying vector space of 120587 The shorthand 119866-representation will be usedextensively

Definition 35 A morphism of 119866-representations 120593 ∶ 1198811113568 rarr 1198811113569 is understood to be continuous homo-morphism of topological vector spaces such that 120593(1198921199071113568) = 119892120593(1199071113568) for all 119892 isin 119866 and 1199071113568 isin 1198811113568 Thisturns the collection of all 119866-representations into a category 119866-Rep so one can talk about isomorphismsautomorphisms etc in the usual manner The morphisms between 119866-representations are also calledintertwining operators

20

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Definition 36 A subrepresentation of a 119866-representation (119881 120587) is a closed subspace 1198811113567 sub 119881 that isstable under 119866-action A quotient representation is a quotient space = 1198811198811113567 by a subrepresentation1198811113567 endowed with the quotient topology In both cases 1198811113567 and are naturally 119866-representations

The119866-action on the direct sum⨁119894isin119868 119881119894 of119866-representations 119881119894119894isin119868 makes it into a119866-representationThe same holds for more general limminusminusrarr of 119866-representations

Call a 119866-representation 119881 ne 0 simple or irreducible if the only subrepresentations (equivalentlyquotient representations) are 0 and 119881 Call it indecomposable if 119881 = 1198811113568 oplus1198811113569 with subrepresentations1198811113568 1198811113569 sub 119881 implies that either 1198811113568 = 0 or 1198811113569 = 0

One of the basic goals of representation theory is to classify the irreducible 119866-representations undervarious constraintsRemark 37 The category 119866-Rep is additive and ℂ-linear In particular the endomorphisms of a 119866-representation 119881 form a ℂ-algebra Endℂ(119881) We caution the reader that unlike the case of finite-dimensional representations 119866-Rep is not an abelian category already in the case of 119866 = 1 sothat 119866-Rep = TopVect every homomorphism 120593 ∶ 119881 rarr 119882 of topological vector spaces admits kernel120593minus1113568(0) and cokernel 119882im(120593) by taking 120593 with dense image one can show that

ker(120593) = 0 = coker(120593) coim(120593) ≃ 119881120593minusrarr 119882 ≃ im(120593)

so that the canonical morphism coim(120593) rarr im(120593) is not an isomorphism if 120593 is not surjective contra-dicting the axioms for abelian categories Such a 120593 ∶ 119881 rarr 119882 already exists for Banach spaces

Let 119881 be in TopVect with a given homomorphism 120587 ∶ 119866 rarr AutVect(119881) so that 119866 acts linearly onthe left of 119881 Consider the orbit map for every 119907 isin 119881

orb119907 ∶ 119866⟶ 119881119892⟼ 119892119907

The following result gives a convenient means to check the continuity of the action map 119886 ∶ 119866 times119881 rarr 119881corresponding to 120587

Proposition 38 For any 119881 in TopVect and a left linear action of 119866 on 119881 (no continuity condition sofar) denote by 120587(119892) the map 119907 ↦ 119892119907 The following are equivalent

1 the action map 119886 ∶ 119866 times 119881 rarr 119881 is continuous ie (119881 120587) is a 119866-representation

2 the conditions below hold (a) there is a dense subspace 1198811113567 sub 119881 such that orb119907 is continuous forall 119907 isin 1198811113567 and (b) for every compact subset 119870 sub 119866 the family 1114106120587(119892) ∶ 119892 isin 1198701114109 of operators on 119881 isequicontinuous

Proof Suppose that 119886 ∶ 119866 times 119881 rarr 119881 is continuous then orb119907 is continuous for all 119907 isin 1198811113567 ∶= 119881 Let119870 sub 119866 be compact Given a neighborhood 119882 ni 0 in 119881 we have to find another neighborhood 119880 ni 0so that 120587(119892)(119880) sub 119882 for each 119892 isin 119870 By the definition of product topology for each 119892 isin 119870 there is aneighborhood119880 prime

119892times119880119892 ni (119892 0) in 119866times119881 that is contained in 119886minus1113568(119882) There is a finite subset 119879 sub 119870 suchthat 119870 = ⋃119905isin119879 119880

prime119905 it remains to take 119880 ∶= ⋂119905isin119879 119880119905

Conversely assume (a) and (b) consider (119892 119907) isin 119866 times 119881 and let 119882 ni 0 be a neighborhood in 119881 Weseek a neighborhood 119880 prime times 119880 of (119892 119907) such that for all (119892prime 119907prime) isin 119880 prime times 119880 we have

119886(119892prime 119907prime) minus 119886(119892 119907) = 120587(119892prime)119907prime minus 120587(119892)119907 isin 119882

For any such neighborhoods 119880 prime 119880 use density to pick 1199071113567 isin 1198811113567 cap119880 ne empty then write 120587(119892)119907 minus 120587(119892prime)119907prime as

1114100120587(119892)119907 minus 120587(119892)11990711135671114103 + 1114100120587(119892)1199071113567 minus 120587(119892prime)11990711135671114103 + 1114100120587(119892prime)1199071113567 minus 120587(119892prime)119907prime1114103

21

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Assume 119880 prime has compact closure then 120587(119892)119907 minus 120587(119892)1199071113567 and 120587(119892prime)1199071113567 minus 120587(119892prime)119907prime approaches zero when 119880shrinks by applying (b) to 119870 ∶= Once 1199071113567 isin 1198811113567 cap 119880 is chosen we can further shrink 119880 prime to make120587(119892)1199071113567 minus 120587(119892prime)1199071113567 arbitrarily small This concludes the proof

Corollary 39 Let (119881 sdot ) be a Banach space and a left action of 119866 on 119881 (no continuity conditions sofar) Then119881 is a119866-representation if and only if 120587(119892) is continuous for every 119892 and the orbit map 119892 ↦ 119892119907is continuous for every 119907 isin 119881

Proof It suffices to prove the ldquoifrdquo part Take 1198811113567 = 119881 in 38 and check the equicontinuity of 120587(119892) ∶ 119892 isin119870 sub AutTopVect(119881) for every compact 119870 sub 119866 More precisely by the uniform boundedness principleof BanachndashSteinhaus it suffices to notice that the continuity of orbit maps implies

sup119892isin119870

1114062120587(119892)1199071114062 lt +infin

for every 119907 isin 119881 which in turn implies the asserted equicontinuity

Example 310 Suppose that 119883 is a locally compact Hausdorff space with a continuous right 119866-actionEquip 119862(119883) ∶= 119891 ∶ 119883 rarr ℂ continuous with the topology defined by semi-norms sdot infin1113653 ∶= sup1113653 sdot where Ω ranges over compact subsets of 119883 This makes 119862(119883) into a member of TopVect Let 119866 act onthe left of 119862(119883) by

119892119891 ∶= 119892119891 = [119909 ↦ 119891(119909119892)] 119891 isin 119862(119883) 119892 isin 119866

Let us verify that 119862(119883) is a 119866-representation It has been observed that 119892(119892prime119891) = (119892119892prime)119891 Apply 38 with119881 = 1198811113567 as follows

bull For every chosen 119891 isin 119862(119883) we contend that 119892 ↦ 119892119891 is a continuous map from 119866 to 119862(119883)Continuity can be checked just at 119892 = 1 This amounts to showing that for every compact Ω sub 119883and every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 such that sup119909isin1113653 |119891(119909119892) minus 119891(119909)| lt 120598whenever 119892 isin 119880 Indeed for every 119909 isin 119883 there exist open 119880119909 ni 1 and 119882119909 ni 119909 in 119866 and 119883 suchthat

(119892 119910) isin 119880119909 times119882119909 ⟹ |119891(119910119892) minus 119891(119910)| lt 120598

Now take a finite subcover of the open covering 119882119909119909isin1113653 of Ω and take 119880 to be the finite inter-section of the corresponding 119880119909

bull Fix a compact 119870 sub 119866 For every 119892 isin 119870 and every compact Ω sub 119883 we know Ω119870 is compact and

120587(119892)1198911113568 minus 120587(119892)11989111135691113653infin = sup119909isin1113653

|1198911113568(119909119892) minus 1198911113569(119909119892)| le 1198911113568 minus 11989111135691113653119870infin

This shows the equicontinuity of the operators 120587(119892) ∶ 119892 isin 119870

Remark 311 When 119883 is second countable an increasing countable sequence of Ω exhausts 119883 and itsuffices to treat the corresponding semi-norms In this case119862(119883) turns out to be Freacutechet If119883 is compactit suffices to take Ω = 119883 and 119883 is a Banach space

Example 312 The same result in 310 holds for 119866 acting on 119862119888(119883) where

119862119888(119883) = limminusminusrarr1113653119862119888(119883Ω)

is equipped with the inductive limit topology where

bull Ω sub 119883 ranges over the compact subsets and

22

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

bull 119862119888(119883Ω) ∶= 1114106119891 isin 119862119888(119883) ∶ Supp(119891) sub Ω1114109 topologized by the sup-norm sdot infin

Then 119862119888(119883) with 119892119891(119909) = 119891(119909119892) is also a 119866-representation on an LF-space and this coincides with 119862(119883)when 119883 is compact

We check the continuity of 119866 times 119862119888(119883) rarr 119862119888(119883) directly Since 119866 is locally compact it suffices toshow the continuity of 119870 times 119862119888(119883) rarr 119862119888(119883) a given compact subset 119870 sub 119866 By the definition of thetopology of limminusminusrarr we further reduce to the composite

119870 times 119862119888(119883Ω) rarr 119862119888(119883Ω sdot 119870) 119862119888(119883)

whereΩ sub 119883 is any compact The continuity of is evident whilst that of119870times119862119888(119883Ω) rarr 119862119888(119883Ωsdot119870)is straightforward to check as everything is compact

Example 313 Let 119883 be a locally compact Hausdorff space endowed with a continuous right 119866-actionFor simplicity we assume that 119883 admits a 119866-invariant Radon positive measure 120583 this is indeed the casewhen 119883 is a homogeneous 119866-space isomorphic to 119867119866 satisfying 120575119866|119867 = 120575119867 by 152

For every 1 le 119901 le infin we have 119866 acting on the Banach space 119881 ∶= 119871119901(119866) by 119892119891 = 119892119891 ∶ 119909 ↦ 119891(119909119892) asusual This makes sense if 119891 isin 119862119888(119883) and 119892119891119901 = 119891119901 for all 119892 isin 119866 Therefore the action extends toall 119891 isin 119871119901(119883) by an approximation argument

Claim 1198711113569(119883) is a 119866-representation We apply 38 with 1198811113567 = 119862119888(119883) as follows The condition (b)is satisfied since 120587(119892) preserves sdot 119901 hence equicontinuous It remains to check (a) that 119892 ↦ 119892119891 iscontinuous for every given 119891 isin 119862119888(119883) ie for every 120598 gt 0 there exists a neighborhood 119880 ni 1 in 119866 suchthat 119892 isin 119880 ⟹ 119892119891 minus 119891119901 lt 120598

Since Supp(119891) is compact there exists a neighborhood 119880 ni 1 such that

|119891(119909119892) minus 119891(119909)| lt 120598120583(Supp(119891))minus1113568119901 for all 119909

Then 119892119891 minus 119891119901 le 120598 as required note that the argument also works for 119901 = infin and in that case we do notneed measures

Representations on function spaces arising from119866-actions on119883 such as the examples 310 312 and313 are called regular representations The most important case is 119883 = 119866 When 119883 = 119866 is finite werevert to the usual regular representation on the finite-dimensional space ℂ[119866] = Maps(119866 ℂ)

We end these discussions by more terminologies

Definition 314 Let (120587 119881) be a119866-representation If119881 is a Hilbert (resp Banach Freacutechet) space we saythat (120587 119881) is a Hilbert (resp Banach Freacutechet) representation In each case such representations form afull subcategory of 119866-Rep

32 Matrix coefficientsDefinition 315 For (120587 119881) be in 119866-Rep Denote by 119881 lowast ∶= HomTopVect(119881 ℂ) the topological dual of119881 It is customary to write ⟨120582 119907⟩ ∶= 120582(119907) for 120582 isin 119881 lowast 119907 isin 119881 Given 120582 119907 the corresponding matrixcoefficient is the continuous function

119888119907otimes120582 ∶ 119866⟶ ℂ119892⟼ 1113993120582120587(119892)1199071113980

Matrix coefficients is bilinear in 120582 and 119907 which justifies the tensor notation

Remark 316 Suppose 119881 ne 0 The HahnndashBanach theorem implies that for every 119907 ne 0 there exists 120582such that 119888119907otimes120582(1) = ⟨120582 119907⟩ ne 0

23

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

We can let119866 act on the left of119881 lowast by the contragredient 120582 ↦ (119892) ∶= 120582∘120587(119892minus1113568) However the choiceof topological structure on 119881 lowast is a subtle issue and in practice it is necessary to pass to some subspaceof 119881 lowast to obtain interesting continuous representations At present we do not put any extra structure on119881 lowast

A Hilbert representation (120587 119881) is called unitary if 120587(119892) is a unitary operator for all 119892 isin 119866 We willsay more about this in 41Remark 317 By Rieszrsquos theorem the topological dual of a Hilbert space (119881 (sdot|sdot)119881 ) is its Hermitianconjugate it is the same space with the new scalar multiplication (119911 119907) ↦ 119907 for 119911 isin ℂ and becomesa Hilbert space with (119907|119908) = (119908|119907)119881 = (119907|119908)119881 Every isin corresponds to the linear functional119907 ↦ (119907|119908)119881

The contragredient representation is therefore given by

(119892) ∶= lowast120587(119892)minus1113568 ∶ rarr

where lowast(⋯) denotes the Hermitian adjoint It satisfies ((119892)119908|120587(119892)119907) = (119908|119907)When 119866 is unitary the contragredient reduces (119892) = 120587(119892) to emphasize the complex conjugation

we will also write ( ) = ( ) in the unitary case and it is again unitaryTherefore 315 specializes as follows

Definition 318 Let (120587 119881) be a Hilbert representation of 119866 For 119907 isin 119881 and 119908 isin the correspondingmatrix coefficient is the continuous map

119888119907otimes119908 ∶ 119866⟶ ℂ119892⟼ (120587(119892)119907|119908)119881

where (sdot|sdot) is the Hermitian form on 119881

We record some basic properties of matrix coefficients

Lemma 319 Let (120587 119881) be a Hilbert representation of 119866 The matrix coefficients map 119908otimes119907 ↦ 119888119907otimes119908 isbilinear in 119881 times Moreover

119888119908otimes119907(119887minus1113568119909119886) = 119888120587(119886)119907otimes(119887)119908(119909) (119886 119887) isin 119866 times 119866 119909 isin 119866 (31)

119888119907otimes119908(119892) = 119888119908otimes119907(119892minus1113568) 119907 119908 isin 119881 119892 isin 119866 if 120587 is unitary (32)

Here we write for the representation 120587 of 119866 on emphasizing that 119888119907otimes119908 is conjugate-linear in 119908 Itis also additive under orthogonal direct sum 1198811113568 oplus 1198811113569 of Hilbert representations namely

119888(119907119907prime)otimes(119908119908prime)(119892) = 119888119907otimes119908(119892) + 119888119907primeotimes119908prime(119892) (33)

with 119892 isin 119866 119907 119908 isin 1198811113568 and 119907prime 119908prime isin 1198811113569

Proof When 120587 is unitary the identity (120587(119892)119908|119907) = (119908|120587(119892minus1113568)119907) = (120587(119892minus1113568)119907|119908) proves (32) Next

1114100120587(ℎminus1113568)120587(119909)120587(119892)119907|1199081114103 = (120587(119909)120587(119892)119907|(ℎ)119908)

translates into (31)

24

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

33 Vector-valued integralsVector-valued integrals are immensely useful for studying continuous representations We shall considera weak version thereof called weak integrals or GelfandndashPettis integrals The canon is surely Bourbaki[3] shorter surveys include [8 326 mdash 329] and [4]

Definition 320 Let 119883 be a locally compact Hausdorff space and 120583 a positive Radon measure thereonLet 119881 be in TopVect and 119891 ∶ 119883 rarr 119881 is a function such that 120582 ∘ 119891 ∶ 119883 rarr ℂ is 120583-measurable for all 120582in the continuous dual 119881 lowast Suppose that 120582 ∘ 119891 is 120583-integrable for all 120582 define the integral int

119883119891 d120583 as the

element120582⟼1114009

1198831113993120582 119891(119909)1113980 d120583(119909) 120582 isin 119881 lowast

in HomVect(119881 lowast ℂ) ie the algebraic dual of 119881 lowast

Since we are working with locally convex Hausdorff spaces 119881 lowast separates points on 119881 by the HahnndashBanach theorem and the natural map 119881 rarr HomVect(119881 lowast ℂ) is injective Therefore it makes sense to askif int

119883119891 d120583 exists inside 119881 which will then take a definite value called the weak integral of 119891 d120583

When119881 is a Banach space the Bochner or the ldquostrongrdquo integrals are instances of weak integrals Dueto its ldquoweakrdquo definition GelfandndashPettis integrals satisfy agreeable functorial properties as illustratedbelow

Proposition 321 Let 119879 ∶ 119881 rarr 119882 be a morphism in TopVect Ifint119883119891 d120583 in the situation of 320 exists

then so does int119883119879119891 d120583 and we have int

119883119879119891 d120583 = 119879 int

119883119891 d120583 Here we identify 119879 with the natural linear

map HomVect(119881 lowast ℂ) rarr HomVect(119882 lowast ℂ) it induces

Proof By definition 119879 int119883119891 d120583 is the linear functional on119882 lowast mapping each 120582 isin 119882 lowast to 1113994120582 ∘ 119879int119883 119891 d1205831113981

which is nothing but int1198831113993120582 1198791198911113980 d120583 Also 1113993120582 1198791198911113980 = 1113993120582 ∘ 119879 1198911113980 is 120583-integrable by assumptions

A Radon measure on 119883 is called bounded if the integration functional 119868 ∶ 119862119888(119883) rarr ℂ is continuousfor the 119871infin-norm A similarly recipe defines the locally bounded measures on 119883 Denote the support ofa measure 120583 by Supp(120583)

Theorem 322 Suppose that120583 is bounded in the situation of 320 If 119891(Supp(120583)) is contained in a convexbalanced bounded and complete subset 119861 sub 119881 then 120582 ∘ 119891 is 120583-integrable for all 120582 isin 119881 lowast and

1114009119883119891 d120583 isin 120583(119883) sdot 119861 sub 119881

Proof This is in [3 sect12 Proposition 8]

Remark 323 Most often we encounter the case of

bull 119891 ∶ 119883 rarr 119881 is continuous

bull 120583 has compact support

In particular 119891(Supp(120583)) is compact The premises in 322 are met whenever compact subsets in 119881have compact convex hulls The latter property holds for quasi-complete spaces A topological vectorspace is called quasi-complete if every bounded closed subset is complete Completeness implies quasi-completeness as expected In particular 322 applies to Freacutechet spaces

Another example of quasi-complete spaces are the continuous duals of barreled spaces endowedwith the topology of pointwise convergence For instance Freacutechet spaces and LF spaces are all barreled

25

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Proposition 324 In the circumstance of 322 suppose that sdot ∶ 119881 rarr ℝge1113567 is a continuous semi-normon 119881 Then

11140651114009119883119891 d1205831114065 le 1114009

119883119891 d120583

Proof Take 119907 = int119883119891 d120583 isin 119881 There exists 120582 isin 119881 lowast with ⟨120582 119907⟩ = 119907 and | ⟨120582 119908⟩ | le 119908 for all 119908 isin 119881

by HahnndashBanach theorem Insert this 120582 into the characterization of int119883119891 d120583

34 Action of convolution algebraIn what follows we will often write the integration of a measure 119891 on a locally compact Hausdorff space119883 as int

119909isin119883119891(119909) omitting the useless d119909 It is convenient to adopt the notations for functions given an

isomorphism Φ ∶ 119884 rarr 119883 write 119891 ∘ Φ for the measure transported to 119884 so that int119884119891 ∘ Φ = int

119883119891 holds

for tautological reasonsLet 119866 be a locally compact group Define the following space of Radon measures on 119866

ℳ (119866) ∶= bounded Radon measures on 119866 ℳ119888(119866) ∶= 1114106 119891 isin ℳ (119866) ∶ Supp(119891) is compact1114109 ℳ (119866) ∶= 1114106 119891 isin ℳ (119866) ∶ 119891 ≪ any right Haar measure1114109 ℳ119888(119866) ∶= ℳ (119866) capℳ119888(119866)

If a right Haar measure 120583 on 119866 is chosen then 1198711113568(119866) d120583 = ℳ (119866)Recall that a subset of 119866 is called symmetric if it is invariant under 119892 ↦ 119892minus1113568 Likewise we say a

function or a measure on 119866 is symmetric if it has the same invariance A topological group always has asymmetric neighborhood basis at 1 replacing each neighborhood 119880 ni 1 by 119880 cap119880minus1113568 ni 1 suffices

Definition 325 An approximate identity is a symmetric neighborhood basis 120081 at 1 isin 119866 together with afamily 120593119880 isin ℳ119888(119866) where 119880 ranges over 120081 such that for all 119880 isin 120081

bull 120593119880 is positive of the form 120601119880 d120583 where 120601119880 isin 119862119888(119866) and 120583 is a right Haar measure

bull int119866120593119880 = 1

bull Supp(120593119880 ) sub 119880 120593119880 (119892) = 120593119880 (119892minus1113568)

When 120081 is countable an approximate identity can be conveniently described as a sequence 1205931113568 1205931113569 hellip inℳ119888(119866) with each 120593119894 symmetric and positive and Supp(120593119894) shrinks to 1 as 120081 does

Proposition 326 For every symmetric neighborhood basis 120081 at 1 isin 119866 there exists approximate iden-tities supported on 120081

Proof Given 119880 isin 120081 use Urysohnrsquos lemma to construct 120593119880 We can force 120593119880 (119892) = 120593119880 (119892minus1113568) by aver-aging

Definition 327 Let (120587 119881) be a continuous representation of 119866 on a quasi-complete space Let ℳ119888(119866)act on the left of 119881 by 119881-valued integrals

120587(120593)119907 = 1114009119866120593(119892)120587(119892)119907 119907 isin 119881 120593 isin ℳ119888(119866)

whose existence is guaranteed by 322

26

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

One should imagine that 120587(119892) = 120587(120575119892) where 120575119892 stands for the Dirac measure at 119892 In general

120587(ℎminus1113568)120587(119891)120587(119892minus1113568) = 120587 1114100ℎ1198911198921114103 ℎ119891119892(119909) = 119891(ℎ119909119892) (34)

where the leftright translation should be understood in terms of measures It preserves the spacesℳ(119866)etc

Lemma 328 This makes 119881 into an ℳ119888(119866)-algebra namely

120587(120593) isin Endℂ(119881)120587(1198861205931113568 + 1198871205931113569)119907 = 119886120587(1205931113568)119907 + 119887120587(1205931113569)119907

120587(1205931113568 ⋆ 1205931113569) = 120587(1205931113568)120587(1205931113569)119907

for all 1205931113568 1205931113569 isin ℳ119888(119866) and 119886 119887 isin ℂ Furthermore if (120593119880 )119880isin120081 is an approximate identity then

lim119880isin120081

120587(120593119880 )119907 = 119907 119907 isin 119881

in the sense of filter bases

Proof The linearity is clear As for convolutions take any 120582 isin 119881 lowast and verify that

1113993120582 120587(1205931113568 ⋆ 1205931113569)1199071113980 = 1114009119892isin119866

1114009ℎisin119867

1205931113568(ℎminus1113568)1205931113569(ℎ119892) 1113993120582 120587(119892)1199071113980

(∵ Fubinirsquos theorem) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113995120582 11141021114009119892isin119866

120587(ℎminus1113568)1205931113569(ℎ119892)120587(ℎ119892)11990711141051113982

(∵ 321) = 1114009ℎisin119867

1205931113568(ℎminus1113568) 1113994120582 120587(ℎminus1113568) 1114100120587(1205931113569)11990711141031113981

= 1113993120582 120587(1205931113568)120587(1205931113569)1199071113980

For the convergence it suffices to show for every continuous semi-norm sdot that

lim119880120587(120593119880 )119907 minus 119907 = 0

since 119881 is locally convex By 324 we have

120587(120593119880 )119907 minus 119907 = 11140651114009119866120593119880 (119892)(120587(119892)119907 minus 119907)1114065

le 1114009119866120593119880 (119892)120587(119892)119907 minus 119907

Given 120598 gt 0 for sufficiently small 119880 we have 120587(119892)119907 minus 119907 lt 120598 when 119892 isin 119880 Hence the last expression isbounded by 120598int

119866120593119880 = 120598

Lemma 329 The subspace 120587 1114100 ℳ119888(119866)1114103119881 of 119881 is 119866-stable and dense In fact the subspace spanned by120587(120593119880 )119881 is already dense when 120593119880 runs over an approximate identity (recall 325)

Proof Apply 328 Note that ℳ119888(119866) is stable under (34) thus 120587 1114100 ℳ119888(119866)1114103119881 is 119866-stable as well

27

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

35 Smooth vectors archimedean caseLet 119866 be a locally compact group and (120587 119881) be a 119866-representation The orbit map orb119907 ∶ 119892 ↦ 119892119907 canalso be used to define representation-theoretic properties of vectors 119907 isin 119881

Definition 330 Let P stand for a property of continuous functions 119866 rarr 119881 For a 119866-representation 119881and 119907 isin 119866 we say that 119907 has the property P if orb119907 ∶ 119866 rarr 119881 has

Since 119907 ↦ orb119907 is a ℂ-linear map from 119881 to Maps(119866119881) if the functions with property P form avector space so are the vectors with property P

In particular when 119866 is an analytic Lie group over a local field 119865 we can talk about 119862infin (infinitelydifferentiable) and119862120596 (analytic) vectors in a119866-representation thereby bringing the smooth structure intothe picture

Definition 331 The 119862infin-vectors in 119881 are called smooth vectors They form a subspace

119881infin ∶= 119907 isin 119881 ∶ smooth vector

In the next few sections we will mainly be interested in the case of archimedean 119865 The same definitionpertains to non-archimedean case as well (see 138) but it has a quite different flavor Note that forarchimedean 119865 one can actually talk about 119862119896-vectors where 119896 isin 0 1 hellip infin120596

In what follows we consider only 119865 = ℝ The case of ℂ follows by ldquoWeil restrictionrdquo namely onecan view a ℂ-group as an ℝ-group by dropping its complex structure

For a Lie group 119866 over 119865 = ℝ let 120100 ∶= Lie(119866) otimesℝ ℂ denote its complexified Lie algebra and let119880(120100) designate the universal enveloping algebra of 120100

Definition 332 Suppose that 119866 is a Lie group over ℝ Let (120587 119881) be in 119866-Rep Define the eponymousaction 120587 of the Lie algebra 120100 by

120587(119883)119907 ∶= dd119905 |119905=1113567

120587(119892(119905))119907 = limℎrarr1113567

120587(119892(ℎ))119907 minus 120587(119892(0))119907ℎ 119883 isin 120100

for every 119907 isin 119881infin where 119892 is any differentiable curve in 119866 with 119892(0) = 1 and 119892prime(0) = 119907 it is customaryto take 119892(119905) = exp(119905119883)

Lemma 333 The operators 120587(119883) above yield a homomorphism of ℂ-algebras

119880(120100) rarr Endℂ(119881infin)

Proof In other words (119883 119907) ↦ 120587(119883)119907 is a representation of the Lie algebra 120100 It boils down to checkthat 120587(119883)(120587(119884)119907) minus 120587(119884)(120587(119883)119907) = 120587([119883 119884])119907 for all 119907 isin 119881infin and 119883119884 isin 120100 This reduces the to basicfact that

exp(119905119883) exp(119904119884) = exp 1114101119905119883 + 119904119884 + 1199041199052 [119883 119884] + higher1114104

[ TO BE CONTINUED ]

4 Unitary representation theoryFix a locally compact group 119866

28

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

41 GeneralitiesRecall that a pre-Hilbert space is a ℂ-vector space 119881 equipped with a positive-definite Hermitian form(sdot|sdot) = (sdot|sdot)119881 if 119881 is complete with respect to the norm 119907 ∶= (119907|119907)11135681113569 we say 119881 is a Hilbert space

Definition 41 A unitary representation (resp pre-unitary representation) of 119866 is a Hilbert (resp pre-Hilbert) space (119881 (sdot|sdot)) together with a linear left 119866-action on 119881 such that

bull for every 119907 isin 119881 the map 119892 ↦ 119892119907 from 119866 to 119881 is continuous

bull for each 119892 isin 119866 the operator 120587(119892) ∶ 119881 rarr 119881 from the 119866-action is unitary ie 120587(119892)(119907) = 119907 forall 119907 isin 119881

Remark 42 In view of 39 with 1198811113567 = 119881 a unitary representation is automatically continuous and is thesame as a Hilbert representation (119881 120587) of 119866 such that all 120587(119892) are unitary operators

On the other hand pre-unitary representations can be completed to unitary ones

Lemma 43 If 119881 is a pre-unitary representation then the Hilbert space completion of 119881 with respectto sdot can be endowed with an induced 119866-action and becomes a unitary representation

Proof For every 119892 isin 119866 the operator 120587(119892) extends uniquely to a unitary operator on The relation120587(119892)120587(119892prime) = 120587(119892119892prime) also extends to It remains to check the continuity of 119892 ↦ 120587(119892)119907 where 119907 isin isfixed To see this take 1199071113567 isin 119881 and note that

120587(119892)119907 minus 120587(119892prime)119907 le 120587(119892)119907 minus 120587(119892)1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 120587(119892prime)1199071113567 minus 120587(119892prime)119907= 119907 minus 1199071113567 + 120587(119892)1199071113567 minus 120587(119892prime)1199071113567 + 1199071113567 minus 119907

We conclude from the continuity of 119892 ↦ 120587(119892)1199071113567 by taking 1199071113567 minus 119907 sufficiently small

Definition 44 A morphism or an intertwining operator 120593 ∶ 1198811113568 rarr 1198811113569 between unitary representationsis a ℂ-linear homomorphism 120593 such that

bull 120593 is unitary ie preserves inner products (119907|119908)1198811113578 = (120593(119907)|120593(119908))1198811113579 for all 119907 119908 isin 1198811113568

bull 120593 respects 119866-actions 119892120593(119907) = 120593(119892119907) for all 119907 isin 1198811113568 and 119892 isin 119866

This turns the collection of all unitary representations into a category which is a non-full subcate-gory of Hilbert representations Isomorphisms between unitary representations are also called unitaryequivalences

Example 45 Let 119866 acts continuously on a locally compact Hausdorff space for simplicity assumethat 119883 carries a 119866-invariant positive Radon measure It is shown in 313 with 119901 = 2 that 1198711113569(119866) is a119866-representation It is actually unitary since we have pointed out in 313 that 119891 ↦ 119892119891 preserves the1198711113569-norm sdot 1113569

The finite direct sum ⨁119899119894=1113568119881119894 of unitary representations 1198811113568 hellip 119881119899 is defined so that the underlying

Hilbert space is the orthogonal direct sum of 1198811113568 hellip 119881119899 The case of countably infinite sum of 1198811113568 helliprequires more care we have to use the completed direct sum ⨁119894ge1113568119881119894 which is the completion of thepre-Hilbert space ⨁119894ge1113568119881119894

The notion of subrepresentations and irreducibility of unitary 119866-representations are the same as thecase of continuous representations in general On the other hand quotients are unnecessary as shown bythe following complete reducibility

29

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Lemma 46 Let 119882 be a subrepresentation of a unitary 119866-representation 119881 Then

119881 = 119882 oplus119882⟂

as unitary representations where 119882⟂ ∶= 119907 isin 119881 ∶ forall119908 isin 119882 (119907|119908) = 0

Proof The orthogonal decomposition surely works on the level of Hilbert spaces It remains to noticethat 119882⟂ is stable under 119866-action

In contrast to the case of modules 46 does not imply that 119881 can be written as a direct sum ofirreducibles In fact in many cases 119881 has no simple submodules as we shall see in 411

The following is a unitary variant of the familiar Schurrsquos Lemma for irreducible representations offinite groups

Theorem 47 Let (120587 119881) be an irreducible unitary representation Then

End119866-Rep(119881) = ℂ sdot id119881

Proof Let 119879 ∶ 119881 rarr 119881 be continuous and linear Suppose that 119879 commutes with 119866-action then so is itsadjoint lowast119879 since

(119908|119892lowast119879119907) = (119879119892minus1113568119908|119907) = (119892minus1113568119879119908|119907) = (119879119908|119892119907) = (119908|lowast119879119892119907)for all 119908 isin 119881 and 119892 isin 119866 Write

119879 = 119879 + lowast1198792 + radicminus1 sdot

119879 minus lowast1198792radicminus1

Therefore to show that 119879 is a scalar we may suppose lowast119879 = 119879 Now let 120590(119879) sub ℝ be the spectrum of 119879 Apply the spectral decomposition for self-adjoint bounded

operators (eg [8 1223 Theorem]) to express 119879 via its spectral measure 119864

119879 = 1114009120590(119879)

120582 d119864(120582)

Here 119864 can be thought as a projection-valued measure on the Borel subsets of 120590(119879) it is canonicallyassociated to 119879 The irreducibility of 119881 together with the 119866-equivariance of 119879 imply that 119864(119860 cap 120590(119879))is either 0 or id119881 for all Borel subsets 119860 sub ℝ ie 120590(119879) is an atom in measure-theoretic terms It followsthat there exists a point 120582 with 119864(120582) = id119881 Indeed take any interval 1198681113568 with spectral measure id119881 bisect it and pass to the one with spectral measure id119881 named 1198681113569 and so forth to obtain nested intervals1198681113568 sup 1198681113569 sup ⋯ such that⋂119894ge1113568 119868119894 = 120582 for some 120582 The zero-one dichotomy then implies that 119864 is supportedat 120582 Therefore 119879 = 120582 sdot id119881

Corollary 48 A unitary representation (120587 119881) is irreducible if and only if End119866-Rep(119881) = ℂ sdot id119881

Proof The ldquoonly ifrdquo part follows from 47 Conversely if 119882 sub 119881 is a proper subrepresentation then119881 = 119882 oplus119882⟂ by 46 hence

End119866-Rep(119881) sup End119866-Rep(119882) oplus End119866-Rep(119882⟂)

has dimension ge 2

Another important consequence of Schurrsquos Lemma is the existence of central characters in the unitarycase

Theorem 49 Let (120587 119881) be an irreducible unitary representation There is a continuous homomorphism120596120587 ∶ 119885119866 rarr 1201381113568 where 119885119866 denotes the center of 119866 such that 120587(119911) = 120596120587(119911)id119881 for every 119911 isin 119885119866 We call120596120587 the central character of 120587

30

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Proof Central elements act on 119881 by scalar multiplication since they commute with 119866 This yields thehomomorphism 120596120587 whose continuity and unitarity follow from that of (120587 119881)

In practice it is convenient to allow any closed subgroup 119885 of 119885119866 By abuse of language we alsosay that (120587 119881) has central character 120596 on 119885 if 119885 acts via the continuous homomorphism 120596 ∶ 119885 rarr 1201381113568

Corollary 410 If 119866 is commutative the irreducible unitary representations are precisely the one-dimensional unitary representations

Proof For commutative 119866 we have 119866 = 119885119866 acts via 120596120587 on irreducible (120587 119881) Conversely one-dimensional representations are clearly irreducible

Example 411 Consider the unitary ℝ-representation 1198711113569(ℝ) It has no irreducible subrepresentationsIndeed an irreducible subrepresentation must be generated by an 1198711113569-function 119891 with 119891(119909+119905) = 120594(119905)119891(119909)where 120594 is a continuous homomorphism ℝ rarr 1201381113568 The only such 1198711113569-function is 0

The next enhancement of 47 will be needed

Theorem 412 Let (120587 119881) and (120590119882) be unitary representations of 119866 and assume that (120587 119881) is irre-ducible Let 119879 be a 119866-equivariant linear map from a dense 119866-stable vector subspace 1198811113567 sub 119881 to 119882 Ifthe graph of 119879

Γ119879 ∶= (119907 119879119907) isin 1198811113567 times119882 ∶ 119907 isin 1198811113567

is closed in 119881 oplus119882 then 1198811113567 = 119881 and 119879 is a scalar multiple of a morphism between unitary representa-tions

Proof We shall make use of basic facts on unbounded operators from Hilbert spaces119881 to119882 in generalsee for example [8 Chapter 13] Such an operator 119879 is only defined on some subspace 119967(119879) of 119881 andcontinuity is not presumed When taking their composites etc domains must shrink suitably In ourcase 119967(119879) = 1198811113567 is dense

The first result is the existence of an adjoint lowast119879 defined on

119967(lowast119879) ∶= 1114106119908 isin 119882 ∶ (119879(sdot)|119908)119882 isin HomTopVect(1198811113567 ℂ)1114109

uniquely determined by (119879119907|119908)119882 = (119907|lowast119879119908)119881 Secondly assume that Γ119879 is closed then 119967(lowast119879) is alsodense then consider the unbounded operator 119876 ∶= 1 + lowast119879119879 from 119881 to itself According to [8 Theorem1313]

bull 119967(119876) = 119907 isin 119967(119879) ∶ 119879119907 isin 119967(lowast119879) maps onto 119881 under 119876

bull there is a continuous endomorphism 119861 of 119881 such that im(119861) sub 119967(119876) and 119876119861 = id119881 In fact119887 ∶= 119861119907 (for all 119907 isin 119881) is characterized in 119882 oplus119881 by

(0 119907) = (119888 lowast119879119888) + (minus119879119887 119887) 119887 isin 119967 (119879) 119888 isin 119967 (lowast119879) (41)

Now apply these results to our setting The domain119967(lowast119879) is a 119866-stable subspace as 1198811113567 = 119967(119879) is Onereadily checks that lowast119879 ∶ 119967 (lowast119879) rarr 119881 is also119866-equivariant From the characterization (41) we concludethat 119861 is 119866-equivariant as well hence 119861 = 120582 sdot id119881 for some 120582 isin ℂtimes by 47 as 119876119861 = id119881

Consequently 119967(119876) = 119967(119879) = 119881 and 119876 = 120582minus1113568id119967(119876) This also implies 119967(lowast119879) sup im(119879) Now

(119879119907|119879119907prime)119881 = (lowast119879119879119907|119907prime)119881 = (120582minus1113568 minus 1)(119907|119907prime)119881 119907 119907prime isin 1198811113567

hence 119879 is a scalar multiple of an isometry 119881 rarr 119882

31

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

42 External tensor productsThe theory of completed tensor products for general locally convex spaces 1198811113568 1198811113569 is delicate For ourpresent purposes the case of Hilbert spaces suffices

Definition 413 Let (1198811113568 (sdot|sdot)1113568) and (1198811113569 (sdot|sdot)1113569) be Hilbert spaces Write 1198811113568 otimes1198811113569 ∶= 1198811113568 otimesℂ 1198811113569 and equipit with an Hermitian form by requiring

(1199071113568 otimes 1199071113569 | 1199081113568 otimes 1199081113569) = (1199071113568|1199081113568)1113568 sdot (1199071113569|1199081113569)1113569 1199071113568 1199081113568 isin 1198811113568 1199071113569 1199081113569 isin 1198811113569

and extend by sesquilinearity to all1198811113568otimes1198811113569 This is well-defined and positive-definite Denote by1198811113568otimes1198811113569the corresponding completion of 1198811113568 otimes 1198811113569

The operation otimes is functorial in 1198811113568 and 1198811113569 turning the collection of Hilbert spaces into a symmetricmonoidal category

Now suppose that 119881119894 is the underlying space of a unitary representations 120587119894 of a locally compactgroup 119866119894 for 119894 = 1 2 Let 1198661113568 times 1198661113569 act on 1198811113568 otimes 1198811113569 via 1205871113568 otimes 1205871113569 this extends to an action on 1198811113568otimes1198811113569by unitary operators In order to stress that it is acted upon by 1198661113568 times 1198661113569 it is customary to designate theresulting representation by 1205871113568 ⊠ 1205871113569 Summing up we obtain

11141001198811113568otimes1198811113569 1205871113568 ⊠ 12058711135691114103 ∶ unitary representation of 1198661113568 times 1198661113569

Proposition 414 If 1205871113568 and 1205871113569 are both irreducible then 1205871113568 ⊠ 1205871113569 is irreducible as a unitary represen-tation of 1198661113568 times 1198661113569

Proof In view of 48 it suffices to show that every continuous 1198661113568 times 1198661113569-equivariant endomorphism 119879of 1205871113568 ⊠ 1205871113569 is 119888 sdot id for some 119888 isin ℂ Since this property can be tested by taking inner product with allelements from 1198811113568 otimes 1198811113569 which are dense in 1198811113568otimes1198811113569 it is equivalent to that

1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

for all 1199071113568 1199071113569 and 1199081113568 1199081113569To achieve this first fix 1199071113569 1199081113569 isin 1198811113569 and define the endomorphism

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198811113568 otimes ℂ = 11988111135681199071113568 ⟼1114100id1198811113578 otimes (sdot|1199081113569)11988111135791114103 (119879(1199071113568 otimes 1199071113569) )

here we view (sdot|1199081113569)1198811113579 as 1198811113569 rarr ℂ It is continuous and 119866-equivariant hence 47 implies 12059311990811135791199071113579 =11988611990811135791199071113579 sdot id1198811113578 for some scalar 11988611990811135791199071113579 Now we have a continuous linear map

119860 ∶ 1198811113569 ⟶ (1198811113569)lowast

1199081113569 ⟼11140941199071113569 ↦ 119886119908111357911990711135791114097

Recall that (1198811113569)lowast ≃ 1198811113569 via the Hermitian form A careful verification reveals that 119860 is 1198661113569-equivarianthence 119860 = 119888 sdot id1198811113579 for some 119888 isin ℂ In other words 11988611990811135791199071113579 = 119888(1199071113569|1199081113569)1198811113579 for all 1199071113569 1199081113569 isin 1198811113569 This implies1114100119879(1199071113568 otimes 1199071113569) | 1199081113568 otimes 11990811135691114103 = 119888(1199071113568|1199081113568)1198811113578 sdot (1199071113569|1199081113569)1198811113579

Proposition 415 Let (120587119894 119881119894) and (120590119894119882119894) be irreducible unitary representations of 119866119894 for 119894 = 1 2 If1205871113568 ⊠ 1205901113568 ≃ 1205871113569 ⊠ 1205901113569 then 120587119894 ≃ 120590119894 for 119894 = 1 2

32

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Proof Suppose that 119879 ∶ 1205871113568⊠1205901113568 ≃ 1205871113569⊠1205901113569 is nonzero Then there must exist 1199071113569 isin 1198811113569 and 1199081113569 isin 1198821113569 suchthat

12059311990811135791199071113579 ∶ 1198811113568 ⟶1198821113568 otimes ℂ = 1198821113568

1199071113568 ⟼1114100id1198821113578 otimes (sdot|1199081113569)11988211135791114103 (119879(1199071113568 otimes 1199071113569))

is nonzero As in the proof of 414 it is1198661113568-equivariant and continuous hence1205871113568 ≃ 1205901113568 by 412 Switchingthe roles of 1198661113568 1198661113569 yields 1205871113569 ≃ 1205871113568

Of particular importance is the case 1198661113568 = 119866 = 1198661113569 and (1198811113568 1198811113569) = (119881 ) where (120587 119881) is a unitaryrepresentation of 119866 and ( ) is its Hermitian conjugate see 317 Recall that (119909|119910) ∶= (119910|119909)119881

By linear algebra there is a ℂ-linear isomorphism

119881 otimes sim⟶ Endfr(119881) ∶= 1114108119879 ∶ 119881ℂrarr 119881 ∶ finite rank continuous1114111

119907 otimes 119908⟼ 119860119907otimes119908 ∶= (sdot|119908)119907(42)

Equip Endfr(119881) with the HilbertndashSchmidt Hermitian inner product which is positive-definite

(119860|119861)HS ∶= Tr(lowast119861 sdot 119860)

It is routine yet amusing to verify that lowast119860119908otimes119907 = 119860119907otimes119908 Hence

(119860119907otimes119908|119860119907primeotimes119908prime)HS= (119907prime|119907)119881 sdot (119908|119908prime)119881 = (119907|119907prime)119881 sdot (119908|119908prime)

which equals the Hermitian inner product in 119881 otimes Also recall that the space of HilbertndashSchmidt operators 119881 rarr 119881 is defined as

HS(119881) ∶= completion of Endfr(119881) relative to (sdot|sdot)HS

The group 119866 times 119866 acts on Endfr(119881) and HS(119881) by

(119886 119887)119879 = 120587(119886)119879120587(119887)minus1113568

This is visibly unitary and corresponds to the action on 119881 otimes under (42)

Theorem 416 The identification (42) extends to an isomorphism

119881 ⊠ ≃ HS(119881)

between unitary 119866 times 119866-representations

Proof The identification is a 119866 times 119866-equivariant isometry between 119881 otimes and Endfr(119881) Now pass tocompletion

43 Square-integrable representationsFix a closed subgroup 119885 sub 119885119866 Central characters of irreducible unitary representations will be definedrelative to 119885 In view of 152 up to ℝtimes

gt1113567 there exists a unique right Haar measure on 119866119885 since thecondition 120575119866|119885 = 120575119885 is trivially satisfied both are trivial from the very definition of 127

As unitary representations are a special kind of Hilbert representations it makes sense to considerthe matrix coefficients 119888119907otimes119908(119892) = (120587(119892)119907|119908)119881 for 119907 otimes 119908 isin 119881 otimes as in 318 If (120587 119881) admits a centralcharacter 120596120587 as in 49 for example when (120587 119881) is irreducible then

119888119907otimes119908(119911119892) = 120596120587(119911)119888119907otimes119908(119892) 119911 isin 119885 119892 isin 119866

In this case |119888119908otimes119907| factors through 119866119885

33

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Definition 417 Denote by 1198711113569(119866119885) the 1198711113569-space defined relative to any right Haar measure on 119866119885More precisely for every continuous homomorphism 120596 ∶ 119885 rarr 1201381113568 set

1198711113569(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885|119891| isin 1198711113569(119866119885) 1114111

It is a Hilbert space under 1198911113569 = int119866119885

|119891|1113569 d120583 once a right Haar measure 120583 is chosen Specifically onestarts from measurable 119891 and take the quotient by functions with sdot = 0 and so on Likewise we definethe other function spaces on 119866 with (120596 119885)-equivariance such as

119862119888(119866119885120596) ∶= 1114108 119891 ∶ 119866 rarr ℂ continuous 119891(119911119909) = 120596(119911)119891(119909) 119911 isin 119885Supp(119891)119885 is compact 1114111

Notice that 1198711113569(119866119885120596) is also a unitary representation of 119866 under 119892 ∶ 119891(119909) ↦ 119891(119909119892) as in 45

Definition 418 Suppose 119866 is unimodular An irreducible unitary representation (120587 119881) of 119866 is calledsquare-integrable or a discrete series representation if the matrix coefficient 119888119907otimes119908 is nonzero and lies in1198711113569(119866119885120596120587) for some 119908 119907

Such representations are also called essentially square-integrable in the literature while the termsquare-integrable is reserved to those with 119888119907otimes119908 isin 1198711113569(119866) ie for 119885 = 1 Square-integrable represen-tations in the latter strict sense exists only when 119885119866 is compact cf 121 and the integration formula in152 when 119885119866 is compact both definitions coincide

In 420 we will extend this definition to non-unimodular groups

Lemma 419 Let (120587 119881) be an irreducible unitary representation (120587 119881) of a unimodular group 119866 Thefollowing are equivalent

1 (120587 119881) is square-integrable

2 119888119907otimes119908 isin 1198711113569(119866119885120596120587) for all 119907 119908 ne 0 and if 119879 ∶ 119907prime ↦ 119888119907primeotimes119908 is not identically zero 119879 is a 119866-equivariant linear map from 119881 into 1198711113569(119866119885120596120587) with closed image which is a scalar multiple ofan isometry

3 there exists an embedding 119881 1198711113569(119866119885120596120587) as unitary 119866-representations

Proof We begin with (1) ⟹ (2) Fix 119908 119907 isin 119881 with 119888119907otimes119908 isin 1198711113569(119866119885120596120587) nonzero We begin byshowing that 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587) for all 119907prime Set

1198811113567 ∶= 1114106119907prime isin 119881 ∶ 119888119907primeotimes119908 isin 1198711113569(119866119885120596120587)1114109

This is a 119866-stable vector subspace by (31) it contains 119907 hence we infer from the irreducibility of 119881 that1198811113567 is dense Define a linear map

119879 ∶ 1198811113567 rarr 1198711113569(119866119885120596120587) 119907prime ↦ 119888119907primeotimes119908

By (31) we see 119879 is 119866-equivariant Equip 1198811113567 with the Hermitian pairing

((119907prime1113568|119907prime1113569)) = (119907prime1113568|119907prime1113569)119881 + (119879119907prime1113568|119879119907prime1113569)1198711113579

We claim that 1198811113567 is a Hilbert space under ((sdot|sdot)) and the graph Γ119879 is closed in 1198811113567 times 1198711113569(119866119885120596120587)Indeed ((sdot|sdot)) is evidently Hermitian and positive definite we set out to show its completeness The

choice of ((sdot|sdot)) implies that for every Cauchy sequence (119907prime119894 )infin119894=1113568 in1198811113567 there exists (119907prime 119888) isin 119881times1198711113569(119866119885120596120587)such that 119907prime119894 rarr 119907prime in 119881 and 119879119907prime119894 rarr 119888 in 1198711113569(119866119885120596120587) Observe that 119907prime119894 rarr 119907prime implies that

|119888119907prime119894otimes119908(119909) minus 119888119907primeotimes119908(119909)| = |(120587(119909)(119907prime119894 minus 119907prime)|119908)| le 119907prime119894 minus 119907prime sdot 119908

34

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

so that 119879119907prime119894 rarr 119888119907primeotimes119908 uniformly on119866 On the other hand one can pass to a subsequence to ensure 119879119907prime119894 rarr 119888pointwise and ae Therefore 119888119907primeotimes119908 = 119888 so that 119907prime isin 1198811113567 This proves both the completeness and closednessparts of our claim

By 412 the claim implies that1198811113567 = 119881 and 119879 is a scalar multiple of an isometry Such maps betweenHilbert spaces always have closed images

To conclude (2) we must also show that 119888119907otimes119908prime isin 1198711113569(119866119885120596120587) for all 119908prime isin 119881 Via (32) and theunimodularity of 119866119885 this is subsumed into the previous case

It is immediate that (2) ⟹ (3) 119879 furnishes the required embedding(3) ⟹ (1) Assume that (120587 119881) is a unitary subrepresentation of 1198711113569(119866119885120596120587) and 119907 isin 119881 is nonzero

There exists isin 1198711113569(119866119885120596120587) such that (119907|)1198711113579(119866119885120596120587) ne 0 We may even assume that isin 119862119888(119866119885120596120587)by density Then 119907prime ↦ (119907prime|)1198711113579(119866119885120596120587) restricts to a continuous linear functional on 119881 of the form119907prime ↦ (119907prime|119908)119881 for a unique 119908 isin 119881 Thus 119888119907otimes119908(1) ne 0 and it remains to show 119888119907otimes119908 isin 1198711113569(119866119885120596120587)

Denote by 120583 the chosen right Haar measure on 119866119885 Put 119906(119909) ∶= (119909minus1113568) to express 119888119907otimes119908(119892) =(120587(119892)119907|)1198711113579(119866119885120596120587) as

1114009119866119885

119906(119909minus1113568)119907(119909119892) d120583(119909)

Its absolute value is bounded by (|119906| ⋆ |119907|)(119892) (convolution on 119866119885) As |119906| isin 119862119888(119866119885) and |119907| isin 1198711113569(119866119885)it follows either from a direct analysis or Youngrsquos inequality 24 with (119901 119902 119903) = (1 2 2) that |119906| ⋆ |119907| isin1198711113569(119866119885) as required

The third condition in 419 makes sense for any locally compact 119866 which leads to the followinggeneral definition

Definition 420 Let 119866 be a locally compact group An irreducible unitary representation (120587 119881) of 119866 iscalled square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen-tation of 1198711113569(119866119885120596120587)

Theorem 421 (Schur orthogonality relation) Let (120587 119881) and (120590119882) be square-integrable representa-tions of a unimodular group 119866 with 120596120587 = 120596120590 and fix a right Haar measure 120583 on 119866119885

1 If (120587 119881) and (120590119882) are not isomorphic as unitary representations then

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 0

for all 119907 119907prime isin 119881 and 119908119908prime isin 119882

2 In the case 120587 = 120590 there exists 119889(120587) isin ℝgt1113567 inverse-proportional to the choice of 120583 such that

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 119889(120587)minus1113568(119907|119908)119881 (119908prime|119907prime)119881

for all 119907 119907prime 119908 119908prime isin 119881

Proof Write 119879 ∶ 119907 ↦ 119888119907otimes119907prime and 119878 ∶ 119908 ↦ 119888119908otimes119908prime They are scalar multiples of isometries and are119866-equivariant Now

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = (119879119907|119878119908)1198711113579(119866119885120596120587) = (lowast119878119879119907|119908)119882

But lowast119878 is also continuous and 119866-equivariant thus 119860 ∶= lowast119878119879 ∶ 119881 rarr 119882 is a morphism in 119866-Rep Then412 implies that 119860 is a scalar multiple of a morphism between unitary representations If (120587 119881) and(120590119882) are non-isomorphic the only possibility is 119860 = 0 and the first item is proved

35

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Next assume 120587 = 120590 Then 119860 = 120582119907prime119908prime sdot id119881 for some 120582119907prime119908prime isin ℂ This amounts to

1114009119866119885

119888119907otimes119907prime119888119908otimes119908prime d120583 = 120582119907prime119908prime(119907|119908)119881

Switching 119907 harr 119907prime and 119908 harr 119908prime replaces the matrix coefficients by their complex conjugates As 119866119885 isunimodular we deduce

120582119907prime119908prime(119908|119907)119881 = 120582119907119908(119907prime|119908prime)119881 Fix 119907 = 119908 ne 0 in the equation above to see that 120582119907prime119908prime = 119888(119908prime|119907prime)119881 for some constant 119888 depending solelyon (120587 119881) and 120583 it is proportional to 120583 Setting 119907 = 119908 = 119908prime = 119907prime to deduce 119888119907otimes11990711135691198711113579(119866119885120596120587) = 119888(119907|119907)

1113571119881

As 119888119907otimes119907(1) ne 0 whenever 119907 ne 0 we see 119888 gt 0 Now set 119889(120587) = 119888minus1113568 to conclude

Corollary 422 The convolutions of square-integrable matrix coefficients on 119866119885 are well-defined Infact for irreducible unitary representations (120587 119881) (120590119882) we have

119888119907otimes119907prime ⋆ 119888119908otimes119908prime(119909) ∶= 1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(119907|119908prime)119888119907primeotimes119908(119909) 120587 = 1205900 120587 ≄ 120590

for all 119907prime 119907 isin 119881 and 119908119908prime isin 119882

Proof Use 32 31 and apply 421 to deduce

1114009119866119885

119888119907otimes119907prime(119892minus1113568)119888119908otimes119908prime(119892119909) d120583(119892) = 1114009119866119885

119888120587prime(119909)119908otimes119908prime(119892)119888119907primeotimes119907(119892) d120583(119892)

=⎧⎪⎨⎪⎩119889(120587)minus1113568(120587prime(119909)119908|119907prime)(119907|119908prime) 120587 = 1205900 120587 ≄ 120590

In the first case it equals 119889(120587)minus1113568(119907|119908prime)119888119908otimes119907prime(119909)

Now enters the bilateral translation on 119866 Observe that 119866 times 119866 acts on the right of 119866 by (119886 119887) ∶ 119909 ↦119887minus1113568119909119886 It acts on the left of functions 119866 rarr ℂ by

(119886 119887)119891 ∶ 119909 ↦ 119891(119887minus1113568119909119886) 119909 isin 119866

Proposition 423 The action ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) makes 1198711113569(119866119885120596) into a unitary representation of119866 times 119866

Proof Since 119866119885 is assumed to be unimodular this action is unitary in other words 119866 carries a 119866 times119866-invariant positive Radon measure The rest of the verification is akin to 45

Corollary 424 Let 120587 be a square-integrable unitary representation of119866 with central character120596 on 119885Then 119907otimes119908 ↦ 119889(120587)minus1113568119888119907otimes119908 yields an isomorphism from120587⊠ onto the irreducible119866times119866-subrepresentationof 1198711113569(119866119885119866 120596) generated by all matrix coefficients of 120587

Proof The map is 119866 times 119866-equivariant by (31) on 119881 otimes It is an isometry on 119881 otimes by 421 since

(119907 otimes 119907prime|119908 otimes 119908prime)119881otimes = (119907|119908)119881 (119907prime|119908prime) = (119907|119908)119881 (119908prime|119907prime)119881

These properties extends to 119881otimes by density The image of any isometry is closed Irreducibility of120587 ⊠ follows from 414

Definition 425 The positive constant 119889(120587) is called the formal degree of 120587 It depends only on theisomorphism class of 120587 and 119866 120583 It is inverse-proportional to the choice of 120583 and generalizes the notionof dimension cf 440

36

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

44 Spectral decomposition the discrete partFix a closed subgroup 119885 sub 119885119866 We still assume 119866119885 unimodular and fix a Haar measure 120583 on it

Definition 426 Let120596 ∶ 119885 rarr 1201381113568 be a continuous character The discrete spectrum with central character120596 is the subrepresentation

1198711113569disc(119866119885120596) ∶= 11140121114106120587 sub 1198711113569(119866119885120596) ∶ irred unitary subrep1114109

By 419 every 120587 in the sum is square-integrable and every square-integrable unitary representation120587 with 120596120587|119885 = 120596 intertwines into 1198711113569disc(119866119885120596) via matrix coefficients

Lemma 427 Every irreducible unitary subrepresentation of 1198711113569(119866119885120596) lies in the closed subspacegenerated by matrix coefficients of square-integrable representations of central character 120596 on 119885

Proof Let 119862 stand for the closed subspace generated by all square-integrable matrix coefficients of cen-tral character 120596 on 119885 If the assertion does not hold some irreducible subrepresentation (120587 119881) wouldhave non-trivial orthogonal projection to 119862⟂ The projection being 119866-equivariant 120587 can be embeddedinside 119862⟂ This is contradictory as explained below

Assume that 119881 sub 119862⟂ and 119891 isin 119881 is nonzero Express 120583 as 120583119866120583119885 according to 153 There exists areal-valued 120593 isin 119862119888(119866) such that the function

Φ(119892) ∶= ( ⋆ 119891)(119892) (continuous by 25)

= 1114009119866119891(ℎ119892)120593(ℎ) d120583119866(ℎ) = 1114009

1198661198851114009119885119891(ℎ119892)120593(119911ℎ)120596(119911) d120583119885(119911) d120583(ℎ)

= 1114009119866119885

119891(ℎ119892)120593120596(ℎ) d120583(ℎ)

is not identically zero where

120593120596(ℎ) ∶= 1114009119885120593(119911ℎ)120596(119911) d120583119885(ℎ) 120593120596 isin 119862119888(119866119885120596)

This is indeed possible by 329 since

Φ(119892) = 1114009119866(ℎ)119891(ℎminus1113568119892) d120583119866(ℎ) = 119871()119891(119892)

where 119871 denotes the continuous 119866-representation 119871(ℎ)119891(119909) = 119891(ℎminus1113568119909) on 1198711113569(119866119885120596)Note thatΦ(119892) = 119888119891otimes119908(119892)where119908 isin 119881 is such that (sdot|120593120596)1198711113579(119866119885120596) = (sdot|119908)119881 We infer that (Φ|119888119891otimes119908) =

Φ11135691198711113579(119866119885120596) ne 0 On the other hand since 119866 is unimodular

(Φ|119888119891otimes119908) =1113962119866119885times119866119885

119891(ℎ119892)120593120596(ℎ) sdot 119888119891otimes119908(119892) d120583(ℎ) d120583(119892)

= 1114009119866119885

11141021114009119866119885

119891(119892)119888119891otimes119908(ℎminus1113568119892) d120583(119892)1114105 120593120596(ℎ) d120583(ℎ)

∵ (31) = 1114009119866119885

1114100 119891 | 119888119891otimes120587(ℎ)11990811141031198711113579(119866119885120596) 120593120596(ℎ) d120583(ℎ)

The inner integral vanishes by assumption Contradiction

In general it may happen that 1198711113569disc(119866119885120596) = 0 for all 120596 See 411

37

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Theorem 428 For each square-integrable unitary representation120587with central character120596 on119885 thereis an isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587HS(119881120587) = 120587

120587 ⊠ sim⟶1198711113569disc(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

bull (120587 119881120587) ranges over the square-integrable representations with central character 120596 on 119885 taken upto isomorphism

bull 119866 times 119866 acts on 1198711113569(119866119885120596) by ((119886 119887)119891)(119909) = 119891(119887minus1113568119909119886) and

bull the ⨁ is a completed orthogonal direct sum of unitary 119866 times 119866-representations

Moreover 120587 ⊠ are pairwise non-isomorphic ie the decomposition above of unitary representationsis multiplicity-free

Proof Combine 424 with 427 Matrix coefficients from different 120587 are orthogonal thus the sum⋯ in426 becomes an orthogonal ⨁ The representations 120587 ⊠ are pairwise distinct by 415 The relationto HS(119881120587) is explicated in 416

Proposition 429 Equip each 119881120587otimes119881120587 (resp HS(119881120587)) in 428 with the ℂ-algebra structure with multi-plication

(119907 otimes 119907prime) ⋆ (119908 otimes 119908prime) ∶= (119907|119908prime)119907prime otimes 119908 (resp 119860 ⋆ 119861 ∶= 119861 ∘ 119860)

Then Coeff restricted to ⨁120587 ⊠ 120587 respects the ⋆-multiplications on both sides

Proof This reduces to 422 and the rule

(sdot|119907prime)119907 ⋆ (sdot|119908prime)119908 = (119907|119908prime)(sdot|119907prime)119908

inside each HS(119881120587) which mirrors (119907 otimes 119907prime) ⋆ (119908 otimes 119908prime)

The inverse of Coeff can be described on the linear span of matrix coefficients as follows

Lemma 430 Let (120587 119881) be a square-integrable unitary representation with central character 120596 on 119885Consider 120593 ∶= Coeff(119907 otimes 119908) isin 1198711113569(119866119885120596) cap 119862(119866119885120596) where 119907 otimes 119908 isin 119881 otimes Then the vector-valuedintegrals

120587()119907prime = 1114009119866119885

(119892)120587(119892)119907 d120583(119892) 119907prime isin 119881

defines an operator 119881 rarr 119881 which equals the (sdot|119908)119907 isin HS(119881) determined by 119907 otimes 119908

Proof We determine 120587()119907prime by applying (sdot|119908prime) isin 119881 lowast where 119908prime isin 119881 is arbitrary By (32) and 421 thiseuqlas

1114009119866119885

(119892)(120587(119892)119907prime|119908prime) d120583(119892) = 119889(120587)1114009119866119885

119888119907otimes119908(119892minus1113568)(120587(119892)119907prime|119908prime) d120583(119892)

= 1114009119866119885

119888119907primeotimes119908prime119888119908otimes119907 d120583 = (119907prime|119908)(119907|119908prime)

Hence 120587()119907prime = (119907prime|119908)119907 and defines a linear endomorphism on 119881 The same description pertains to(sdot|119908)119907

38

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

45 Usage of compact operatorsWe make systematic use of the formalism in 327

Lemma 431 Let 120593 isin ℳ119888(119866) and let (120587 119881) be a unitary representation of 119866 Define (119892) = 120593(119892minus1113568) inthe sense of measures Then

lowast120587(120593) = 120587 11141001114103

In particular if 120593 is real-valued and 120593(119892) = 120593(119892minus1113568) for all 119892 then 120587(120593) = lowast120587(120593)

Proof For all 119907 119908 isin 119881

(120587(120593)119907|119908) = 1114009119866120593(119892)(120587(119892)119907|119908) = 1114009

119866120593(119892)(119907|120587(119892minus1113568)119908)

= 1114009119866(119892)(120587(119892)119908|119907) = 1114009

119866(119892)(120587(119892)119908|119907)

= 1114100120587 11141001114103119908|1199071114103 = 1114100119907|120587 111410011141031199081114103

Example 432 Now consider 119866 acting on 1198711113569(119866119885120596) where 119885 sub 119885119866 is closed and 120596 ∶ 119885 rarr 1201381113568 is achosen continuous homomorphism call this representation 119877 Choose a right Haar measure 120583 on 119866119885with 120583 = 120583119866120583119885 as in 153 Assume that 120593 isin 119862119888(119866) d120583 then we have

1114100119877(120593)1198911114103 (119909) = 1114009119866120593(119892)119891(119909119892) d120583119866(119892)

= 1114009119866119885

119870120593(119909 119910)119891(119910) d120583119866119885(119892)

with119870120593(119909 119910) ∶= 1114009

119885120593(119909minus1113568119911119910)120596(119911) d120583119885(119911) 119909 119910 isin 119866119885 (43)

Clearly 119870120593(119909 119910) ∶ 119866 times 119866 rarr ℂ is continuous and satisfies

119870120593(119909119911prime 119910119911Prime) = 120596(119911prime)120596(119911Prime)minus1113568119870120593(119909 119910) 119911prime 119911Prime isin 119885

It expresses 119877(120593) as an integral transform on 1198711113569(119866119885120596) with kernel 119870120593 This is slightly different fromthe usual picture since a character 120596 intervenes

Lemma 433 (GelfandndashGraevndashPiatetski-Shapiro) Let (120587 119881) be a unitary representation of 119866 Supposethat there exists an approximate identity (120593119880 )119880isin120081 (recall 325) such that each 120587(120593119880 ) ∶ 119881 rarr 119881 is acompact operator Then (120587 119881) is a completed orthogonal sum of its irreducible subrepresentationsMoreover each irreducible constituent appears in this decomposition with finite multiplicity

Proof The following arguments are due to Langlands Set

119982 ∶= sets of mutually orthogonal irreducible subrepresentations sub 120587

Note that119982 is nonempty since empty isin Σ and it is partially ordered by set inclusion Zornrsquos lemma affordsa maximal element 119878 isin 119982 indeed every chain in 119982 has the upper bound furnished by taking unionConsider the subrepresentation 1205871113567 ∶= ⨁120590isin119878120590 of 120587 We claim that 1205871113567 = 120587

Let the subrepresentation 1205871113568 of 120587 be the orthogonal complement of 1205871113567 We will derive a contradic-tion from 1205871113568 ne 0 by exhibiting an irreducible subrepresentation inside 1205871113568 which would violate themaximality of 119878

39

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Denote by 1198811113568 sub 119881 the underlying space of 1205871113568 By 328 there exists 120593119880 from the approximateidentity such that 119879 ∶= 120587(120593119880 )|1198811113578 ne 0 Note that 119879 is a self-adjoint compact operator since 120587(120593119880 ) isby 325 and 431 The spectral theorem for such operators (eg [8 Theorems 425 + 1223]) implies that119879 has an eigenvalue 120582 isin ℝ ∖ 0 and the eigenspace 119881119879=120582

1113568 is nonzero and finite-dimensional Take asubspace 119881

1113568 sub 119881119879=1205821113568 of minimal dimension subject to the condition that 119881

1113568 = 119882119879=120582 = 119882 cap 119881119879=1205821113568 for

some 119866-stable closed subspace 119882 sub 1198811113568 Among these 119882 their intersection 119882min ∶= ⋂119882 ∶ 119882119879=120582 =1198811113568 is the smallest one We contend that 119882min is irreducible Otherwise there would be an orthogonal

decomposition 119882min = 119860 oplus 119861 into nonzero 119866-stable closed subspaces and

1198811113568 = 119882119879=120582

min = 119860119879=120582 oplus 119861119879=120582

The minimality assumption on 1198811113568 implies that 119881

1113568 equals 119860119879=120582 or 119861119879=120582 but this contradicts the mini-mality of 119882min

We conclude that 120587 = ⨁120590isin119878120590 Suppose that some irreducible unitary representation 1205901113567 (consideredup to isomorphism) intervenes with multiplicity119898 Then for every120593119880 in the approximate identity120587(120593119880 )will restrict to 119898-copies of 1205901113567(120593119880 ) By assumption we may choose 120593119880 such that 1205901113567(120593119880 ) ne 0 thus has anonzero eigenvalue 120582 As the 120582-eigenspace of 120587(120593119880 ) is finite-dimensional119898must be finite as well

Proposition 434 In the setting of 432 suppose that 119866119885 is compact then 1198711113569(119866119885120596) decomposes intoa completed orthogonal direct sum of irreducible unitary representations each constituent appearingwith finite multiplicity

In particular 1198711113569(119866119885120596) = 1198711113569disc(119866119885120596) when 119866 is unimodular

Proof In view of 433 it suffices to show that each120593 isin 119862119888(119866) acts on 1198711113569(119866119885120596) via compact operatorsIn fact they act as HilbertndashSchmidt operators a standard result asserts that integral operators on 1198711113569(119883)prescribed by a kernel 119870 isin 1198711113569(119883 times 119883) are HilbertndashSchmidt If 119883 is compact then every continuous119870 ∶ 119883 times 119883 rarr ℂ is 1198711113569 One can take 119883 = 119866119885 if 120596 = 1 but the arguments with 120596 ∶ 119885 rarr 1201381113568 and 119870120593satisfying (43) are entirely analogous

46 The case of compact groupsSuppose that 119866 is a unimodular group and 119866119885 is compact Therefore all irreducible unitary representa-tions of 119866 are square-integrable relative to 119885 Fix a Haar measure 120583 on 119866119885

Proposition 435 Every irreducible unitary representation (120587 119881) of 119866 is finite-dimensional

Proof The argument below is credited to L Nachbin Take a nonzero 119907 isin 119881 Define for all 119908 isin 119881 thevector-valued integral

119860119908 ∶= 1114009119866119885(119908|120587(119892)119907)119881 sdot 120587(119892)119907 d120583(119892)

By 324 we know 119860119908119881 le 119908119881 sdot 1199071113569119881 sdot 120583(119866119885) and 119860 is easily seen to be 119866-equivariant as 119866 actsunitarily Hence 119860 = 120582 sdot id119881 for some 120582 isin ℂ by 47 moreover (119860119907|119907)119881 = int |(119907|120587(119892)119907)|1113569 d120583(119892) implies120582 gt 0 Now let1198811113567 sub 119881 be any finite-dimensional subspace with the corresponding orthogonal projection119864 ∶ 119881 rarr 1198811113567 sub 119881 Consider 119864119860 restricted to 1198811113567 to obtain

1114009119866119885

1114100 sdot |119864120587(119892)1199071114103 sdot 119864120587(119892)119907 d120583(119892) = 120582 sdot id1198811113577 ∶ 1198811113567 rarr 1198811113567

Inside Endℂ(1198811113567) we take traces on both sides and pass it under the integral sign to deduce

120583(119866119885)1199071113569119881 ge 1114009119866119885

119864120587(119892)1199071113569 d120583(119892) = 120582 dim1198811113567

This bound shows that dim119881 is finite

40

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Theorem 436 (PeterndashWeyl the 1198711113569 version) Let 120596 ∶ 119885 rarr 1201381113568 be a continuous homomorphism There isan isomorphism of unitary 119866 times 119866-representations

Coeff ∶ 120587∶irred

HS(119881120587) = 120587∶irred

120587 ⊠ sim⟶1198711113569(119866119885120596)

(119907120587 otimes 119908120587)120587 ⟼1114012120587119889(120587)119888119907120587otimes119908120587

and the decomposition is multiplicity-free Consequently the matrix coefficients of irreducible unitaryrepresentations with central character 120596 span a dense subspace in 1198711113569(119866119885120596)

Furthermore the inverse of Coeff is determined as follows Let 120593 be a matrix coefficient of 120587 then

Coeffminus1113568(120593) = 120587() ∶= 1114009119866119885

(119892)120587(119892) d120583(119892) isin Endℂ(119881120587)

Proof To obtain the decomposition simply combine 428 with 434 The inverse map follows from430

Remark 437 In deriving 436 we did not use the fact that each irreducible 120587 has finite dimension Infact we can deduce 435 from 436 as follows Restrict 119866times119866-representations to 119866times 1 ≃ 119866 so that thespectral decomposition degenerates into an isomorphism of unitary 119866-representations

120587120587otimes119881120587 ≃ 1198711113569(119866119885120596)

where119881120587 still carries its Hilbert space structure but now with trivial119866-action By 434 we know 120587mustoccur in 1198711113569(119866119885120596) with finite multiplicity which is exactly dim119881120587

Corollary 438 The space 1198711113569(119866119885120596) is closed under convolution⋆ Specifically if each119881120587otimes119881120587 (respHS(119881120587)) is endowed with the ℂ-algebra structure from 429 then Coeff is an isomorphism between ℂ-algebras with multiplication given by ⋆ on both sides

Proof Since 1198711113569(119866119885120596) sub 1198711113568(119866119885120596) by compactness 1198711113569-property is preserved under convolution by23 It suffices to check the compatibility with convolution on a dense subspace of ⨁120587120587 ⊠ and 429will do the job

Finer properties of the spectral decomposition will be studied in the section on direct integrals

47 Basic character theoryWe keep the assumptions from sect46 119866 is a locally compact group 119885 sub 119885119866 is closed and119866119885 comes witha chosen Haar measure 120583 Fix a continuous homomorphism 120596 ∶ 119885119866 rarr 1201381113568 Throughout this subsectionwe choose the Haar measure normalized by

120583(119866119885) = 1

For endomorphisms of finite-dimensional vector spaces it is safe to take trace The following is thusjustified

Definition 439 The character of a finite-dimensional unitary representation (120587 119881) of 119866 is the contin-uous function

Θ120587 ∶ 119866⟶ ℂ119892⟼ Tr 1114100120587(119892) ∶ 119881 rarr 1198811114103

41

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

We clearly haveΘ120587oplus120590 = Θ120587+Θ120590 andΘ = Θ120587 Note thatΘ120587 does not involve choices of measures

Proposition 440 The formal degree of an irreducible unitary representation (120587 119881) equals dim119881

Proof Choose an orthonormal basis 1199071113568 hellip 119907119889 of 119881 Then 120587(119892)1199071113568 hellip 120587(119892)119907119889 form an orthonormal basisfor every 119892 isin 119866 Then 1114100119888119907119894otimes119907119895(119892)11141031113568le119894119895le119889 is a transition matrix between orthonormal bases hence unitary

It follows that sum119894119895 |119888119907119894otimes119907119895(119892)|1113569 = 119889 Integrate both sides over 119866119885 and apply 421 to see sum119894119895 119889(120587)minus1113568 = 119889

that is 119889(120587) = 119889

Theorem 441 If 1199071113568 hellip 119907119889 is an orthonormal basis of an irreducible unitary representation (120587 119881) of119866 then

Θ120587(119892) =1198891114012119894=1113568119888119907119894otimes119907119894(119892) isin 119862(119866119885119866 120596120587)

Furthermore for any two irreducible unitary representations 120587 120590 of 119866 with central character 120596 on 119885

(Θ120587|Θ120590)1198711113579(119866119885120596) =⎧⎪⎨⎪⎩1 120587 ≃ 1205900 120587 ≄ 120590

Proof For any linear endomorphism 119860 ∶ 119881 rarr 119881 we have Tr(119860) = sum119889119894=1113568(119860119907119894|119907119894) This proves the first

assertion The second assertion stems from Schurrsquos orthogonality relations 421 and 440

Theorem 442 For the adjoint action 119891(119909)119892↦ 119891(119892minus1113568119909119892) of 119866 on 1198711113569(119866119885120596) the invariant-subspace is

1198711113569(119866119885120596)119866 = 120587∶irred120596120587|119885=120596

ℂΘ120587

Proof We know that 120587⊠ = HS(119881120587) = Endℂ(119881120587) as119866times119866-representations by 416 The adjoint action119891(119909) ↦ 119891(119892minus1113568119909119892)mirrors the diagonal119866-action on120587⊠ which in turn corresponds to119860 ↦ 120587(119892)119860120587(119892)minus1113568on Endℂ(119881120587) topology does not matter here since dim119881120587 lt +infin

Hence the space of 119866-invariants matches End119866(120587) = ℂ sdot id by 412 In fact id corresponds tosum119894 119907119894 otimes 119907119894 isin 119881120587 otimes 119881120587 where 1199071113568 1199071113569 hellip is any orthonormal basis it has norm 119889(120587) and maps to 119889(120587)Θ120587 isin1198711113569(119866119885120596) under Coeff

As in the case of finite groups characters distinguish representations

Proposition 443 Let 120587 120590 be finite-dimensional unitary representations Then 120587 ≃ 120590 if and only ifΘ120587 = Θ120590

Proof Decompose 120587 and 120590 into orthogonal sums of irreducibles which is possible by 46 and thenapply 441

5 Unitary representation of compact Lie groups (a sketch)Throughout this section Lie groups are always taken over ℝ For a Lie group 119866 we denote by 1201001113567 its realLie algebra and 120100 ∶= 1201001113567 otimesℝ ℂ Denote the center of 1201001113567 as 1201191113567 Below we write 119894 = radicminus1

42

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

51 Review of basic notionsLet 119866 be a connected compact Lie group Below is an incomplete collection of basic results on thestructure of such groups The reader may consult any available text on compact Lie groups for detailssuch as [5]

bull The exponential map exp ∶ 1201001113567 rarr 119866 is surjective

bull The Lie algebra is reductive namely 1201001113567 = 1201191113567 oplus [1201001113567 1201001113567] and [1201001113567 1201001113567] is semisimple

bull The Lie subalgebra 1201191113567 corresponds to the identity connected component (119885119866)∘ of 119885119866 and [1201001113567 1201001113567]corresponds to a closed Lie subgroup119866119904119904 We have119866 = (119885119866)∘ sdot119866119904119904 In particular 119866119904119904 is a compactsemisimple Lie group

bull A torus means a connected commutative compact Lie group Therefore the group 119885∘119866 above is atorus

bull A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering whichis a connected compact and simply connected Lie group Applied to 119866119904119904 there is a finite coveringof 119866 of the form

119885 times 119866sc ↠ 119866 (51)

where 119885 is a torus and 119866sc is a connected and simply connected compact Lie group

Let 119879 be a torus Since exp ∶ 120113 rarr 119879 is an open surjective homomorphism between Lie groups wesee 1201131113567ℒ ≃ 119879 where ℒ ∶= ker(exp) is a lattice in 1201131113567 Consequently 119879 ≃ (ℝℤ)dim119879 as Lie groupsDefine the character lattice as

119883lowast(119879) ∶= Hom(119879 1201381113568) 119894120113lowast1113567 sub 120113lowast

Here Hom is taken in the category of Lie groups and the inclusion is given by mapping 120594 ∶ 119879 rarr 1201381113568 tothe 120582 isin 119894120113lowast1113567 such that

120594(exp(119883)) = exp(⟨120582 119883⟩) 119883 isin 1201131113567so 120582 is essentially the derivative of 120594 at 1 This identifies 119883lowast(119879) with a lattice in 119894120113lowast1113567 namely

1114106120582 isin 119894120113lowast1113567 ∶ 120582(ℒ ) sub 2120587119894ℤ1114109

and corresponds to minus120582 whenever 120594 corresponds to 120582 It is sometimes beneficial to read 120594 as ldquo119890120582rdquoNow we consider maximal tori in 119866 they are closed tori 119879 sub 119866 maximal with respect to inclusion

bull The Lie algebra 1201131113567 of a maximal torus 119879 is an abelian subalgebra in 1201001113567 and vice versa

bull The maximal tori in 119866 are all conjugate

bull Given a maximal torus 119879 every element is conjugate to some element of 119879 If 119905 119905prime isin 119879 areconjugate in 119866 then they are actually conjugate in the normalizer 119873119866(119879)

bull 119885119866 is contained in the intersection of all maximal tori

Fix a maximal torus 119879 sub 119866 there is a decomposition

120100 = 120113 oplus 1114008120572isin1113662(120100120113)

120100120572

into 119879-eigenspaces under the adjoint action Ad ∶ 119879 rarr Endℂ(120100) HereΦ = Φ(120100 120113) sub 119894120113lowast1113567 is the set of rootsfor (120100 120113) all the subspaces 120100120572 are 1-dimensional This yields a reduced root system on 1198941201131113567 By choosinga system of positive roots we decompose Φ = Φ+ ⊔ (minusΦ+)

43

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Given any root 120572 the associated coroot is denoted by isin 1198941201131113567The positive coroots cut out the ldquoacute Weyl chamberrdquo

119966+ ∶= 1113998120572isin1113662+

ge 0 sub 119883lowast(119879) otimes ℝ = 119894120113lowast1113567

the elements of 119966+ are called dominant Denote 119966 ∘+ ∶= ⋂120572isin1113662+ gt 0

The algebraically defined Weyl group Ω(120100 120113) is generated by root reflections

119904120572 ∶ 120582 ↦ 120582 minus ⟨120582 ⟩ 120572 120572 isin Φ

on 119894120113lowast1113567 It is a finite group It also acts on 1198941201131113567 by duality we require that ⟨119908120582119883⟩ = 1113994120582119908minus11135681198831113981 for all119908 isin Ω(120100 120113)

It turns out thatΩ(120100 120113) together with its action on 119894120113lowast is canonically isomorphic to the analytically de-finedΩ(119866 119879) = 119873119866(119879)119885119866(119879) This is essentially done by reducing to the case 1201001113567 = 120112120114(2) Furthermore119966+ is a fundamental domain for the Ω(119866 119879)-action we have

119894120113lowast1113567 = 1113996119908isin1113653(119866119879)

119908119966+

119908 ne 119908prime ⟹ 119908119966 ∘+ cap 119908prime119966 ∘

+ = empty(120582 isin 119966+ and 119908120582 isin 119966+) ⟹ 119908120582 = 120582

The reflections relative to simple roots in Φ(120100 120113) generate Ω(119866 119879) We write ℓ ∶ 119882 rarr ℤge1113567 for thelength function for Ω(119866 119879) with respect to these generators Recall that (minus1)ℓ(sdot) ∶ Ω(119866 119879) rarr plusmn1 is ahomomorphism of groups

When 119866 is semisimple we define

119875 ∶= 120582 isin 119883lowast(119879) otimes ℝ ∶ forall120572 isin Φ ⟨120582 ⟩ isin ℤ 119876 ∶= 1114012

120572isin1113662+ℤ120572

It is a general fact about root systems that they are both lattices in 119883lowast(119879) otimes ℝ and

119875 sup 119883lowast(119879) sup 119876

If 119866 is simply connected (resp 119885119866 = 1 ie adjoint) then 119875 = 119883lowast(119879) (resp 119876 = 119883lowast(119879)) Forsemisimple 119866 in general 119875119883lowast(119879) ≃ 1205871113568(119866 1) In any case 119875 contains the half-sum of positive roots

120588 ∶= 121114012120572isin1113662+

120572 isin 119875

52 Algebraic preparationsDefinition 51 For any Ω-stable lattice Λ sub 119883lowast(119879) otimes ℝ and any commutative ring 119860 (usually 119860 = ℤ)the elements of the group 119860-algebra 119860[Λ] are expressed uniquely in the exponential notation

1114012120582isin1113657

119888120582119890120582 119888120582 isin 119860 ∶ finite sum

subject to the relation 119890120582119890120583 = 119890120582+120583 and the usual laws of algebra The Weyl groupΩ(119866 119879) acts119860-linearlyby mapping 119890120582 to 119890119908120582

44

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Once an isomorphism Λ ≃ ℤoplus119903 is chosen ℤ[Λ] is identifiable with ℤ[119883plusmn11135681113568 hellip 119883plusmn1113568

119903 ] In particularℤ[Λ] is a unique factorization domain since it is a localization of ℤ[1198831113568 hellip 119883119903] Same for ℂ[119883lowast(119879)]

Fix a maximal torus 119879 sub 119866 together with a system of positive roots Φ+ sub Φ = Φ(120100 120113) We willmainly work inside ℤ[119883lowast(119879)] Keep in mind that 120582 isin 119883lowast(119879) (thus the symbol 119890120582) also signifies thecharacter 119879 rarr 1201381113568 mapping exp(119883) to exp(⟨120582 119883⟩) for all 119883 isin 1201131113567 By linearity we obtain a map

ℂ[119883lowast(119879)]⟶ 119862(119879)

As 119883lowast(119879) sub 119894120113lowast1113567 it is reasonable to define the complex conjugation in ℂ[119883lowast(119879)] by

1114012120582119888120582119890120582 =1114012

120582119888120582119890minus120582

which extends the complex conjugation on Hom(119879 1201381113568) sub ℤ[119883lowast(119879)]

Lemma 52 The map ℂ[119883lowast(119879)] rarr 119862(119879) is an embedding of ℂ-algebras where 119862(119879) is equipped withpointwise addition and multiplication It is Ω(119866 119879)-equivariant and respects complex conjugation onboth sides

Proof For injectivity apply the linear independence of characters to 119879 The other assertions are inherentin the construction

It is convenient to enlarge 119883lowast(119879) to the commensurable lattice Λ ∶= 11135681113569119883

lowast(119879) to accommodate forelements like 1198901205721113569 for 120572 isin Φ Notice that Λprime sub Λ implies ℤ[Λprime] sub ℤ[Λ]

Definition 53 The Weyl denominator is the element

Δ ∶= 119890120588 1114015120572isin1113662+

(1 minus 119890minus120572) = 1114015120572isin1113662+

11141001198901205721113569 minus 119890minus12057211135691114103

which lives in ℤ1114095 11135681113569119883lowast(119879)1114098 or ℤ[119875] if 119866 is semisimple

Several algebraic lemmas are in order

Lemma 54 For every 119908 isin Ω(119866 119879) we have 119908Δ = (minus1)ℓ(119908)Δ

Proof It suffices to check this for 119908 = 119904120573 where 120573 isin Φ+ is a simple root Use Δ = prod120572isin1113662+(1198901205721113569 minus 119890minus1205721113569)and the fact that 119904120573(120572) isin Ψ+ for all 120572 isin Φ+ ∖ 120573 and 119904120573(120573) = minus120573

In what follows it would be convenient to be able to expand (1 minus 119890minus120572)minus1113568 in a formal series sum119896ge1113567 119890minus119896120572for every 120572 isin Φ+ For every lattice Λ sub 119883lowast(119879) otimes ℝ we can rdquocompleterdquo ℤ[Λ] in the direction of theclosed cone minus119966 sub 119883lowast(119879)otimesℝ (the negative ldquoobtuse Weyl chamberrdquo) generated by minus(Φ+) to accommodatesuch formal series Specifically we start with the monoid algebra ℤ[minus119966 cap Λ] take its adic completionwith respect to the interior ideal generated by minus119966 ∘ cap Λ finally invert 119890120582 ∶ 120582 isin Λ to reach the desiredalgebra

Lemma 55 Suppose thatΛ is a lattice in119883lowast(119879)otimesℝ If 120583 120584 isin Λ∖0 are not proportional then 1minus1198901205831 minus 119890120584 do not generate the same ideal in ℤ[Λ]

Proof By working in some ldquocompletionrdquo of ℤ[Λ] as explained earlier we expand

1 minus 1198901205831 minus 119890120584 = (1 minus 119890

120583)1114012119896ge1113567

119890119896120584 =1114012119896ge1113567

119890119896120584 minus1114012119896ge1113567

119890120572+119896120573

If 120572 120573 are linearly independent in119883lowast(119879)otimesℝ there will be infinitely many nonzero terms so 1minus119890120584 cannotdivide 1 minus 119890120583 in ℤ[Λ] The same reasoning applies to 1113568minus119890120584

1113568minus119890120583

45

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Lemma 56 Suppose thatΛ is anΩ(119866 119879)-stable lattice in119883lowast(119879)otimesℝ such that ⟨Λ ⟩ sub ℤ for all 120572 isin ΦAn element 120594 isin ℤ[Λ] satisfies 119908120594 = (minus1)ℓ(119908)120594 for all 119908 isin Ω(119866 119879) if and only if 120594 isin Δ sdot ℤ[Λ]1113653(119866119879)

Proof Given 54 it suffices to prove the ldquoonly ifrdquo direction Using

bull the unique factorization property in ℤ[Λ]

bull 1 minus 119890minus120572 and 1 minus 119890minus120573 do not generate the same ideal since Φ is a reduced root system

bull and the fact that 119890120583 isin ℤ[Λ]times for all 120583

it suffices to show that (1 minus 119890minus120572) ∣ 120594 for all 120572 isin Φ+ Write 120594 = sum120582 119888120582119890120582 By assumption we have

119888119904120572120582 = minus119888120582 120582 isin Λ

and 119904120572(120582) = 120582 minus ⟨120582 ⟩ 120572 Collecting 1 119904120572-orbits we see that 120594 can be expressed as a ℤ-linear sum ofexpressions

119890120582 minus 119890120582minus⟨120582⟩120572 = 119890120582 11141001 minus 119890minus⟨120582⟩1205721114103

The last term is divisible by 1 minus 119890minus120572 as 119896 ∶= ⟨120582 ⟩ isin ℤ handle the cases 119896 ge 0 and 119896 lt 0 separately

Remark 57 There exists a lattice Λ sub 119883lowast(119879) otimes ℝ such that Λ sup 119883lowast(119879) cup 120588 and ⟨Λ ⟩ sub ℤ for all120572 isin Φ For example in the finite covering 120587 ∶ 119885 times 119866sc ↠ 119866 of (51) take a maximal torus in 119885 times 119866scof the form 119885 times 119879sc that surjects onto 119879 and consider Λ ∶= 119883lowast(119885 times 119879sc) Pull-back induces 119883lowast(119879) Λwhereas 119883lowast(119879) otimes ℚ = Λ otimes ℚ On the other hand the roots remain unaltered under pull-back since 120587induces an isomorphism on Lie algebras

Definition 58 Take Λ as in 57 For 120582 isin ℤ[119883lowast(119879)] cap 119966 + define

120594120582 ∶= Δminus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 1114015120572isin1113662+

(1 minus 119890minus120572)minus1113568 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus120588

It is a priori an element in the ring of fractions of ℤ[Λ]

We will show in 510 that 120594120582 is independent of the choice of Φ+

Lemma 59 For every 120582 isin ℤ[119883lowast(119879)] we have 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Proof Fix 120582 Clearly sum119908isin1113653(119866119879)(minus1)ℓ(119908)119890119908(120582+120588) varies by (minus1)ℓ(sdot) under the Ω(119866 119879)-action on ℤ[Λ]hence lies in Δℤ[Λ]1113653(119866119879) by 56 On the other hand the second expression of 120594120582 can be expanded into aformal series by completing ℤ[Λ] (thus ℤ[119883lowast(119879)]) in the direction of the closed cone generated by Φ+

Observe that 119908(120582 + 120588) = 119908120582 + (119908120588 minus 120588) isin 119883lowast(119879) as

119908120588 minus 120588 = minus 1114012120572isin1113662+119908120572notin1113662+

120572 isin 119876

So the expansion of 120594120582 in formal series involves only 119890120583 with 120583 isin 119883lowast(119879) A comparison with the previousstep shows that 120594120582 isin ℤ[119883lowast(119879)]1113653(119866119879)

Remark 510 It is a standard fact that the choices of Φ+ in Φ form a Ω(119866 119879)-torsor under conjugationIf we pass from Φ+ to 119908Φ+ then 120594120582 is also transported by 119908 The result above shows that 120594120582 is actuallyindependent of Φ+

46

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Write 120594120582 = sum120583 119888120583119890120583 and let Supp(120594120582) ∶= 1114106120583 isin 119883lowast(119879) ∶ 119888120583 ne 01114109 As observed above Supp(120594120582) isΩ(119866 119879)-stable

For 120582 120582prime isin 119883lowast(119879) otimes ℝ we write

120582 ≺ 120582prime ⟺ 120582prime minus 120582 = 1114012120572isin1113662+

119899120572ge1113567

120572 (52)

Lemma 511 Let 120582 isin 119883lowast(119879) We have 120582 isin Supp(120594120582) and it appears with coefficient 1 If 120583 isin Supp(120594120582)then 120583 = 120582 minus sum120572isin1113662+ 119899120572120572 for some coefficients 119899120572 isin ℤge1113567 in particular 120583 ≺ 120582

Proof Work in a suitably completed group algebra to write

120594120582 = 119890minus120588 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)

= 119890120582 1114015120572isin1113662+

11141001 + 119890minus120572 + 119890minus1113569120572 +⋯1114103 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908(120582+120588)minus(120582+120588)

The basic theory of root systems says 119908(120582 + 120588) minus (120582 + 120588) is always of the form minussum120572isin1113662+ 119899120572120572 with119899120572 isin ℤge1113567 and is trivial only when 119908 = 1

Lemma 512 Let 120582 120583 isin 119883lowast(119879) cap 119966+ The coefficient of 1 = 1198901113567 in 120594120582Δ120594120583Δ equals |Ω(119866 119879)| (resp 0) if120582 = 120583 (resp 120582 ne 120583)

Proof Look at the coefficient of 1198901113567 in

|Ω(119866 119879)|minus11135681114012119908(minus1)ℓ(119908)119890119908(120582+120588) sdot 1114012

119907(minus1)ℓ(119907)119890minus119907(120583+120588)

= |Ω(119866 119879)|minus11135681114012119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

Suppose that 119908(120588 + 120582) = 119907(120588 + 120583) Since 120588 + 120582 120588 + 120583 isin 119966 ∘+ this implies 119908 = 119907 and 120582 = 120583 Therefore

1198901113567 appears with multiplicity |Ω(119866 119879)| in sum119908119907(minus1)ℓ(119908)+ℓ(119907)119890119908(120582+120588)minus119907(120583+120588)

53 Weyl character formula semisimple caseStill assume 119866 to be a connected compact Lie group with maximal torus 119879 and choose a system ofpositive roots Φ+ sub Φ = Φ(120100 120113) Fix the Haar measures 120583119866 on 119866 and 120583119879 on 119879 both of total mass 1

Notice that since Φ = Φ+ ⊔ (minusΦ+)

ΔΔ =1114015120572isin1113662

(1 minus 119890120572) =1114015120572isin1113662

(119890120572 minus 1) isin ℤ[119883lowast(119879)]

thus defines a continuous function on 119879 We denote this function as |Δ|1113569

Theorem 513 (Weylrsquos integration formula) If 119891 ∶ 119866 rarr ℂ is measurable then

1114009119866119891 d120583119866 =

1|Ω(119866 119879)|1113962(119879119866)times119879

119891(119892minus1113568119905119892)|Δ|1113569(119905) d(120583 times 120583119879 )(119879119892 119905)

where 120583 = 120583119866120583119879 is the measure on 119879119866 prescribed in 153

47

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Proof Note that 120583(119866119879) = 1 by plugging 119891 = 1 into (19) We shall transform the integral by pulling itback through the submersion

119860 ∶ (119879119866) times 119879 ⟶ 119866(119879119892 119909)⟼ 119892minus1113568119909119892

In doing integration we can work on a dense open subset of119866 (resp 119879) Here we consider the regular lo-cus119866reg (resp 119879reg) 119905 isin 119879 lies in 119879reg if and only if119885119866(119905) = 119879 set119866reg = ⋃119892isin119866 119892minus1113568119879reg119892 Once restrictedto 119866reg the map 119860 becomes a Ω(119866 119879)-torsor with the left Ω(119866 119879)-action by (119879119892 119909) ↦ (119879119908119892119908119909119908minus1113568)Indeed if 119892119909119892minus1113568 = ℎ119910ℎminus1113568 isin 119866reg then 119909 is conjugate to 119910 through a unique 119908 isin Ω(119866 119879) with represen-tative isin 119866 Thus minus1113568ℎminus1113568119892 centralizes 119909 so that minus1113568ℎminus1113568119892 isin 119873119866(119879) and has trivial image in Ω(119866 119879)showing that 119892119909119892minus1113568 and ℎ119910ℎminus1113568 differ by a unique 119908 isin Ω(119866 119879)

It remains to pinpoint the Jacobian Choose the invariant volume forms on 1201001113567 1201131113567 and 12010011135671201131113567 whichmatch the volume forms for 120583119866 120583119879 and 120583 at 1 = exp(0) via the exponential Decompose 120100 = 120113 oplus 120109 with120109 = ⨁120572 120100120572 the latter is actually defined over ℝ namely 120109 = 1201091113567 otimes ℂ and 1201091113567 carries the correspondingvolume form

Scrutinize the local behavior of 119860 around a chosen (119879119892 119909)

1201091113567 times 1201131113567 ni (119883 119884) ↦ 119892minus1113568 exp(minus119883) exp(119884)119909 exp(119883)119892

Using the invariance of Haar measures under bilateral translations it suffices to consider

(119883 119884) ↦ exp(minus119883) exp(119884)119909 exp(119883)119909minus1113568

= exp(minus119883) exp(119884) exp(Ad(119909minus1113568)119883)

by writing Ad(119909minus1113568)119883 = 119909minus1113568119883119909 its differential at (0 0) is seen to be

(119883 119884) ↦ 1114100Ad(119909minus1113568) minus 11114103119883 + 119884

Relative to our compatible choice of volume forms the Jacobian factor at (119879119892 119909) is precisely the absolutevalue of det 1114100Ad(119909minus1113568) minus 1|1201091114103 To conclude the computation note that

det 1114100Ad(119909minus1113568) minus 1|1201091114103 =1114015120572isin1113662

(120572(119909)minus1113568 minus 1) = 1114015120572isin1113662+

|120572(119909) minus 1|1113569

here 120572 is the character corresponding to what we denoted by 119890120572

Remark 514 By the basic properties of conjugacy classes on119866 the inclusion 119879 119866 induces a bijection119879119882 rarr 119866conj This induces a matching between invariant functions A conjugation-invariant function119891 ∶ 119866 rarr ℂ is continuous if and only if its Ω(119866 119879)-invariant avatar 119891 = 119891|119879 ∶ 119879 rarr ℂ is We supply alow-tech proof of this fact as follows

The ldquoonly ifrdquo part is obvious Conversely assume 119891 is continuous and consider any sequence 119909119895 rarr 119909in 119866 write 119909119895 = 119892119895119905119895119892minus1113568119895 with 119905119895 isin 119879 Thanks to compactness we may pass to subsequences to assumethat the limits 119892119895 rarr 119892 and 119905119895 rarr 119905 exist thus 119909 = 119892119905119892minus1113568 We deduce that 119891(119909119895) = 119891(119905119895) rarr 119891(119905) = 119891(119909)whence the continuity of 119891

Denote the spaces of conjugation-invariant functions on 119866 as 119862(119866)119866minusinv 1198711113569(119866)119866minusinv and so on

Corollary 515 For 120585 120578 isin ℂ[119883lowast(119879)]1113653(119866119879) we identify them asΩ(119866 119879)-invariant continuous functionson 119879 then viewed as a conjugation-invariant continuous functions on 119866 by 514 Let 1198881113567(120585 120578) be thecoefficient of 1198901113567 = 1 in 120585Δ120578Δ = 120585|Δ|1113569 then

(120585|120578)1198711113579(119866) =1198881113567(120585 120578)|Ω(119866 119879)|

48

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Proof Plug these objects into 513 to see

(120585|120578)1198711113579(119866) =1

|Ω(119866 119879)| 1114009119879120585120578|Δ|1113569 d120583119879

Recall that the integral over 119879 of a character 120594 ∶ 119879 rarr 1201381113568 equals zero (resp 1 = 120583119879 (119879)) when 120594 ne 1(resp 120594 = 1) The assertion becomes visible after expanding 120585Δ120578Δ in ℤ[119883lowast(119879)]

Lemma 516 Assume 119866 is semisimple The elements 120594120582 for 120582 isin 119966 ∘+ cap 119883lowast(119879) form a ℤ-basis for

ℤ[119883lowast(119879)]1113653(119866119879) Moreover in terms of the embedding ℤ[119883lowast(119879)]1113653(119866119879) 119862(119866)119866minusinv of 52 they areorthonormal with respect to (sdot|sdot)1198711113579(119866)

Proof Observe thatℤ[119883lowast(119879)]1113653(119866119879) has aℤ-basis of the form 120585120582 ∶= sum120583isin119882sdot120582 119890120583 where 120582 isin 119966 ∘+cap119883lowast(119879)

Write120594120582 = 1114012

120583isin119966 ∘+cap119883lowast(119879)119888120582120583120585120583 119888120582120583 isin ℤ

Then 511 entails that 119888120582120582 = 1 and 119888120582120583 ne 0 ⟹ 120583 ≺ 120582 Note that the ≺ defined in (52) induces a totalorder on 119883lowast(119879) otimes ℝ since 120100 is semisimple This shows that (119888120582120583)120582120583 is an upper triangular matrix overℤ when 120582 120583 are enumerated along ≺ with diagonal entries equal to 1 Hence it is invertible and thisimplies that 120594120582120582 form a ℤ-basis of ℤ[119883lowast(119879)]1113653(119866119879)

The orthonormal property follows from 512 and 515

Now apply the results in sect46 with 119885 = 1 Given a unitary representation 120587 of 119866 with dim119881120587 lt infinits restriction to 119879 decomposes into irreducibles By 410 this is

120587|119879 = 1114008120583isin119883lowast(119879)

120583oplusmult(120587∶120583)

where mult(120587 ∶ 120583) isin ℤge1113567 stands for the multiplicity of 120583 in 120587|119879 On the other hand its character Θ120587restricted to 119879 gives a finite sum

Θ120587(119905) = 1114012120583isin119883lowast(119879)

mult(120587 ∶ 120583)120583(119905) 119905 isin 119879

This function is Ω(119866 119879)-invariant Therefore Θ120587|119879 actually yields an element of ℤ[119883lowast(119879)]1113653(119866119879) cor-responding to sum120583 mult(120587 ∶ 120583)119890120583 We call those 120583 with nonzero multiplicities as the weights of 119866

The weight-multiplicities mult(120587 ∶ 120583)120583isin119883lowast(119879)cap119966+ determineΘ120587|119879 and thenΘ120587 which in turn deter-mines 120587 up to isomorphism by 443

Theorem 517 (Weyl character formula) Assume 119866 is semisimple Then the set of 120582 isin 119883lowast(119879) cap 119966+is in bijection with the set of isomorphism classes of irreducible unitary representations 120587 of 119866 It isdetermined by

120582 harr 120587 ⟺ Θ120587 = 120594120582where we identify Θ120587 with an element of ℤ[119883lowast(119879)]1113653(119866119879) as above In fact 120582 is the highest weight of 120587relative to Φ+ namely all the other weights are ≺ 120582 see (52) it occurs with multiplicity one in 120587

Proof Given120587 we regardΘ120587 as an element ofℤ[119883lowast(119879)]1113653(119866119879) and expand it intosum120582 119899120582120594120582 with 119899120582 isin ℤusing 516 Since those 120594120582 have been shown to be orthonormal 441 yields

1 = (Θ120587|Θ120587)1198711113579(119866) =11140121205821198991113569120582

49

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

Hence there is exact one nonzero 119899120582 with 119899120582 isin minus1 +1 We must have 119899120582 = +1 so that Θ120587 = 120594120582otherwise by 511 mult(120587 ∶ 120582) = 119899120582 = minus1 would be absurd That result also implies the assertion abouthighest weight

All in all we have an injection 120587 ↦ 120582 by matchingΘ120587 = 120594120582 Since all theΘ120587 form an orthonormalbasis of the Hilbert space 1198711113569(119866)119866minusinv whilst the 120594120582 are also orthonormal by 516 this map must besurjective as well

Proposition 518 (Weyl denominator formula) Assume 119866 is semisimple Then

Δ = 1114012119908isin1113653(119866119879)

(minus1)ℓ(119908)119890119908120588

Proof By considerations of the highest weight the trivial representation of119866must correspond to 120582 = 0The assertion then follows from 1205941113567 = 1

Theorem 519 (Weyl dimension formula) Assume 119866 is semisimple Let 120587 be an irreducible unitaryrepresentation with highest weight 120582 Then

dim119881120587 =prod120572isin1113662+ 1113993120582 + 120588 1113980prod120572isin1113662+ 1113993120588 1113980

Proof The dimension equals Θ120587(1) Write ΔΘ120587 = sum119908(minus1)119908120588119890119908(120582+120588) Take appropriate derivatives

[ NOT FINISHED YET ]

54 Extension to non-semisimple groups[ UNDER CONSTRUCTION ]

References[1] N Bourbaki Eacuteleacutements de matheacutematique Fascicule XXIX Livre VI Inteacutegration Chapitre 7 Mesure

de Haar Chapitre 8 Convolution et repreacutesentations Actualiteacutes Scientifiques et Industrielles No1306 Hermann Paris 1963 222 pp (2 inserts) (cit on pp 7 9)

[2] N Bourbaki Eacuteleacutements de matheacutematique Topologie geacuteneacuterale Chapitres 1 agrave 4 Hermann Paris1971 xv+357 pp (not consecutively paged) (cit on p 8)

[3] N Bourbaki Eacuteleacutements de matheacutematique XXV Premiegravere partie Livre VI Inteacutegration Chapitre 6Inteacutegration vectorielle Actualiteacutes Sci Ind No 1281 Hermann Paris 1959 106 pp (1 insert) (citon p 25)

[4] Paul Garrett ldquoVector-valued integralsrdquo httpwwwmathumnedusimgarrettmfun Apr 21 2014(cit on p 25)

[5] Brian Hall Lie groups Lie algebras and representations Second Vol 222 Graduate Texts inMathematics An elementary introduction Springer Cham 2015 pp xiv+449 isbn 978-3-319-13466-6 978-3-319-13467-3 doi 101007978-3-319-13467-3 url httpdxdoiorg101007978-3-319-13467-3 (cit on p 43)

[6] Edwin Hewitt and Kenneth A Ross Abstract harmonic analysis Vol I Second Vol 115 Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Structureof topological groups integration theory group representations Springer-Verlag Berlin-New York1979 pp ix+519 isbn 3-540-09434-2 (cit on pp 8 9)

50

[7] John M Lee Introduction to smooth manifolds Second Vol 218 Graduate Texts in MathematicsSpringer New York 2013 pp xvi+708 isbn 978-1-4419-9981-8 (cit on p 12)

[8] Walter Rudin Functional analysis Second International Series in Pure and Applied MathematicsMcGraw-Hill Inc New York 1991 pp xviii+424 isbn 0-07-054236-8 (cit on pp 25 30 3140)

[9] Jean-Pierre Serre Lie algebras and Lie groups Vol 1500 Lecture Notes in Mathematics 1964lectures given at Harvard University Corrected fifth printing of the second (1992) edition Springer-Verlag Berlin 2006 pp viii+168 isbn 978-3-540-55008-2 3-540-55008-9 (cit on p 11)

51

• Topological groups
• • Definition of topological groups
  • Local fields
  • Measures and integrals
  • Haar measures
  • The modulus character
  • Interlude on analytic manifolds
  • More examples
  • Homogeneous spaces
  • • Convolution
    • • Convolution of functions
      • Convolution of measures
      • • Continuous representations
        • • General representations
          • Matrix coefficients
          • Vector-valued integrals
          • Action of convolution algebra
          • Smooth vectors archimedean case
          • • Unitary representation theory
            • • Generalities
              • External tensor products
              • Square-integrable representations
              • Spectral decomposition the discrete part
              • Usage of compact operators
              • The case of compact groups
              • Basic character theory
              • • Unitary representation of compact Lie groups (a sketch)
                • • Review of basic notions
                  • Algebraic preparations
                  • Weyl character formula semisimple case
                  • Extension to non-semisimple groups
                  • • References